
Class _JVI_L15_ 

Book. Jz&. 

Gowright^°_ 

COPYRIGHT DEPOSIT. 







^s 



ELEMENTS OF MECHANISM. 



BY 



PETER SCHWAMB, S.B., 

Professor of Machine Design, Massachusetts Institute of Technology, 



AND 



ALLYNE L. MERRILL, S.B., 

Associate Professor of Mechanism, Massachusetts Institute of Technology. 



FIRST EDITION. 
FIRST THOUSAND. 



NEW YORK: 

JOHN WILEY & SONS. 

London : CHAPMAN & HALL, Limited. 

1904. 






LIBRARY of CONGRESS 


Two Copies Received 


NOV 4 I9U4 


^opyriant tntry 



Copyright, 1904, 

BY 

PETER SCHWAMB 

AND 

ALLYNE L. MERRILL. 



0^-31(0°) 



ROBERT DRUMMOND, PRINTER, NEW YORK, 



PEEFACE. 



The main subject-matter of this work was written during 1885 by 
Peter Schwamb and has been used since then, in the form of printed 
notes, at the Massachusetts Institute of Technology, as a basis for in- 
struction in mechanism, being followed by a study of the mechanism of 
machine tools and of cotton machinery. The notes were written because 
a suitable text-book could not be found which would enable the required 
instruction to be given in the time available. They have accomplished 
the desired result, and numerous inquiries have been received for copies 
from various institutions and individuals desiring to use them as text- 
books. This outside demand, coupled with a desire to revise the notes, 
making such changes and additions as experience has proved advisable, 
is the reason for publishing at this time. 

Very little claim is made as to originality of the subject-matter 
which has been so fully covered by previous writers. Such available 
matter has been used as appeared best to accomplish the object desired. 
Claim for consideration rests largely on the manner of presenting the 
subject, which we have endeavored to make systematic, clear, and prac- 
tical. 

Among the works consulted and to which we are indebted for sug- 
gestions and illustrations are the following: " Kinematics of Machin- 
ery" and u Der Konstrukteur," by F. Reuleaux, the former for the 
discussion of linkages, and the latter for various illustrations of mechan- 
isms; " Principles of Mechanism," by S. W. Robinson, for the discussion 
of non-circular wheels; " Kinematics/' by C. W. MacCord, for the dis- 
cussion of annular wheels and screw-gearing; " Machinery and Mill- 
work/' by Rankine; " Elements of Mechanism/' by T. M. Goodeve; 
and "Elements of Machine Design/' by W. C. Unwin. 

Peter Schwamb. 
Allyne L. Merrill. 

October 20, 1904. 

ffi 



CONTENTS. 



CHAPTER I. 

PAGE 

Introduction 1 

CHAPTER II. 

Composition and Resolution of Velocities. — Motions of Rigidly- 
connected Points. — Instantaneous Axis. — Centroids 7 

CHAPTER III. 
Pairs of Elements. — Bearings and Screws. — Worm and Wheel 14 

CHAPTER IV. 

Rolling Cylinders and Cones Connected by Force-closure. m \ 
Rolling of Non-cylindrical Surfaces. — Lobed Wheels 23 

CHAPTER V. 

Connection by Bands or Wrapping Connectors. — Belts, Cords, 
and Chains 40 

CHAPTER VI. 
Levers. — Cams 59 

CHAPTER VII. 

LlNKWORK 73 

CHAPTER VIII. 

Parallel Motions. — Straight-line Motions 114 

v 



Vl CONTENTS. 

CHAPTER IX. 

TAGE 

Intermittent Linkwork. — Intermittent Motion 128 

CHAPTER X. 
Wheels in Trains 151 

CHAPTER XL 
Aggregate Combinations 169 

CHAPTER XII. 

Gearing. — Construction op Gear-teeth 186 

Index 255 



ELEMENTS OF MECHANISM. 



CHAPTER I. 
INTRODUCTION. 

1. The science of Mechanism treats of the designing and construc- 
tion of machinery. 

2. A Machine is a combination of resistant bodies so. arranged that 
by their means the mechanical forces of nature can be compelled to 
produce some effect or work accompanied with certain determinate 
motions.* In general, it may be properly said that a machine is an 
assemblage of moving parts interposed between the source of power 
and the work, for the purpose of adapting the one to the other. 

No machine can move itself, nor can it create motive power; this 
must be derived from external sources, such as the force of gravitation, 
the uncoiling of a spring, or the expansion of steam. As an example of 
a machine commonly met with, an engine might be mentioned. It is 
able to do certain definite work, provided some external force shall 
act upon it, setting the working parts in motion. We shall find that 
it consists of a fixed frame, supporting the moving parts, some of which 
cause the rotation of the engine shaft, others move the valves distrib- 
uting the steam to the cylinder, and still others operate the governor 
which controls the engine. These moving parts will be so arranged 
that they make certain definite motions relative to each other when an 
external force, as steam, is applied to the piston. 

3. The operation of any machine depends upon two things : first, the 
transmission of certain forces, and second, the production of determinate 
motions. In designing, due consideration must be given to both of 
these, so that each part may be adapted to bear the stresses imposed on 
it, as well as have the proper relative motion in regard to the other parts 
of the machine. But the nature of the movements does not depend upon 
the strength or absolute dimensions of the moving parts, as can be shown 

*Reuleaux: Kinematics of Machinery. 



2 INTRODUCTION. 

by models whose dimensions may vary much from those requisite for 
strength, and yet the motions of the parts will be the same as those of 
the machine. Therefore, the force and the motion may be considered 
separately, thus dividing the science of Mechanism into two parts, viz.: 

1° Pure Mechanism, which treats of the motions and forms of the 
parts of a machine, and the manner of supporting and guiding them, 
independent of their strength. 

2° Constructive Mechanism, which involves the calculation of the 
forces acting on different parts of the machine; the selection of mate- 
rials as to strength and durability in order to withstand these forces, 
taking into account the convenience for repairs, and facilities for manu- 
facture. 

In what follows, we shall, in general, confine ourselves to the first part, 
pure mechanism, or what is sometimes called "the geometry of machin- 
ery"; but shall in some cases consider the forces in action. 

Then our definition of a machine might be modified to accord with 
the above, as follows: 

A Machine is an assemblage of moving parts so connected that when 
the first, or recipient, has a certain motion, the parts where the work is 
done, or effect produced, will have certain other definite motions. 

4. A Mechanism is a term applied to a portion of a machine where 
two or more pieces are combined, so that the motion of the first compels 
the motion of the others, according to a law depending on the nature of 
the combination. For example, the combination of a crank and con- 
necting-rod with guides and frame, in a steam engine, serving to con- 
vert reciprocating into circular motion, would thus be called a mechanism. 

The term Elementary Combination is sometimes used synonymously 
with A Mechanism. 

A machine is made up of a series or train of mechanisms. 

5. Motion and Rest are necessarily relative terms within the limits 
of our knowledge. We may conceive a body as fixed in space, but we 
cannot know that there is one so fixed. If two .bodies, both moving in 
space, remain in the same relative position in regard to each other, they 
are said to be at rest, one relatively to the other; if they do not, either 
may be said to be in motion relatively to the other. 

Motion may thus be either relative, or it may be absolute, provided we 
ansume some point as fixed. In what follows, the earth will be assumed 
to be at rest, and all motions referred to it will be considered as absolute. 

Path. — A point moving in space describes a line called its path, 
which may be rectilinear or curvilinear. The motion of a body is 
determined by the paths of three of its points selected at pleasure. If 
the motion is in a plane, two points suffice, and if rectilinear, one point 
suffices to determine the motion. 



CONTINUOUS MOTION. 3 

Direction. — In a given path, a point can move in either of two direc- 
tions only, which may be designated in various ways: as up, + , or 
down, — ; to the right, + , or left, — ; with the clock, + , or the reverse, 
— ; direction, as well as motion, being relative. 

6. Continuous Motion. — When a point goes on moving indefinitely 
in a given path in the same direction, its motion is said to be continuous. 
In this case the path must return on itself, as a circle or other closed curve. 
A wheel turning on its bearings affords an example of this motion. 

7. Reciprocating Motion. — When a point traverses the same path 
and reverses its motion at the ends of such path the motion is said to be 
reciprocating. 

Vibration and Oscillation are terms applied to reciprocating circular 
motion, as that of a pendulum. 

8. Intermittent Motion. — When the motion of a point is interrupted 
by periods of rest, its motion is said to be intermittent. 

9. Revolution and Rotation. — A point is said to revolve about an 
axis when it describes a circle of which the centre is in, and the plane is 
perpendicular to, that axis. When all the points of a body thus move 
the body is said to revolve about the axis. If the axis passes through 
the body, as in the case of a wheel, the word rotation is used synonymously 
with revolution. It frequently occurs that a body not only rotates about 
an axis passing through itself, but also moves in an orbit about another 
axis. In order to make the distinction between the two motions more 
clear, we shall consider the first as a rotation, and the second as a revo- 
lution; just as we say, the earth rotates on its axis and revolves around 
the sun. 

An Axis of Rotation is a line whose direction is not changed by the 
rotation; a fixed axis is one whose position, as well as its direction, 
remains unchanged. 

A Plane of Rotation is a plane perpendicular to the axis of rotation. 

Right-handed Rotation is the same in direction as the motion of the 
hands of a watch, and is generally considered to be positive. Left- 
handed rotation is in the opposite direction and is consequently con- 
sidered as negative. 

10. Cycle of Motions. — When a mechanism is set in motion and its 
parts go through a series of movements which are repeated over and 
over, the relations between and order of the different divisions of the 
series being the same for each repetition, we have in one of these series 
what is called a cycle of motions. For example, one revolution of the 
crank of a steam engine causes a series of different positions of the 
piston-rod, and this series of positions is repeated over and over for each 
revolution of the crank. 



4 INTRODUCTION. 

The Period of a motion is the interval of time elapsing between two 
successive passages of a point through the same position in the same 
direction. 

ii. Driver and Follower. — That piece of a mechanism which is 
supposed to cause motion is called the driver, and the one whose motion 
is effected is called the follower. 

12. Frame. — The frame of a machine is a structure that supports 
the moving parts and regulates the path, or kind of motion, of many of 
them directly. In discussing the motions of the moving parts, it is con- 
venient to refer them to the frame, even though it may have, as in the 
locomotive, a motion of its own. 

13. Velocity. — Velocity is the rate of motion of a point in space. 
When the motion is referred to a point in the path of the body its velocity 
is expressed in linear measure. When the point is rotating continuously, 
or for the instant, about some axis, its motion may be referred to the 
axis when its velocity is expressed in angular measure. In the first 
case it has linear velocity and in the second case angular velocity. 

Velocity is uniform when equal spaces are passed over in equal times, 
however small the intervals into which the time is divided. The veloc- 
ity in this case is the space passed over in a unit of time, and if s repre- 
sent the space passed over in the time t, the velocity v will be 

-I m 

Velocity is variable when unequal spaces are passed over in equal 
intervals of time, increasing spaces giving accelerated motion and decreas- 
ing spaces giving retarded motion. The velocity when variable is the 
limit of the space passed over in a small interval of time, divided by the 
time, when these intervals of time become infinitely small. If As repre- 
sent the space passed over in the time At, then 

As 
v = limit of -7- as At diminishes indefinitely, 

or 

•-* & 

The uniform linear velocity of a point is measured by the number of 
units of linear distance passed over in a unit of time, as feet per minute, 
inches per second, etc. When the velocity is variable it is measured by 
the distance which would be passed over in a unit of time, if the point 
retained throughout that time the velocity which it had at the instant 
considered. 

14. Angular Velocity. — The angular velocity of a point is measured 
by the number of units of angular space which would be swept over in 



MODES OF TRANSMISSION. 5 

a unit of time by a line joining the given point with a point outside of 
its path, about which the angular velocity is desired. The angular 
space is here measured by circular measure, or the ratio of the arc to 
the radius. The unit angle, or radian, is one subtended by an arc equal 
to the radius. The angular velocity of a point is therefore expressed in 
radians per unit of time. (In what follows l.v. will be used to designate 
linear velocity and a.v. angular velocity, for the sake of brevity.) 

Inasmuch as in circular motion the linear velocity of the point is the 
velocity along the arc, we may write: 



l.v. 

a.v. = - 
r 

from which 



radius 7 ^ 



l.v. = a.v. X radius (4) 

Thus when the a.v. remains the^same, the l.v. is directly proportional 
to the radius. For example, given a line of shafting with pulleys of 
various diameters, the a.v. of all the pulleys is the same, while the l.v's 
of points in the rims of the pulleys are directly proportional to the 
respective radii. If N represents the number of revolutions per minute 
(r.p.m.) and R represents the radius of one of the pulleys in feet, we 
have the a.v. equal to 2tzN radians per minute, while the l.v. of the 
rim of the pulley would be 2~NR feet per minute (f.p.m.). 

15. Modes of Transmission. — If we leave out of account the action 
of natural forces of attraction and repulsion, such as magnetism, one 
piece cannot move another, unless the two are in contact or are connected 
to each other by some intervening body that is capable of communicating 
the motion of the one to the other. In the latter case, the motion of 
the connector is usually unimportant, as the action of the combination 
as a whole depends upon the relative motion of the connected pieces. 
Thus motion can be transmitted from driver to follower; 

1° By direct contact. 

2° By intermediate connectors. 

16. Links and Bands.— An intermediate connector can be rigid or 
flexible. When rigid it is called a link, and it can either push or pull, 
such as the connecting-rod of a steam engine. Pivots or other joints 
are necessary to connect the link to the driver and follower. 

If the connector is flexible, it is called a band, which is supposed to 
be inextensible, and only capable of transmitting a pull. A fluid con- 
fined in a suitable receptacle may also serve as a connector, as in the 
hydraulic press. Here we might call the fluid a pressure-organ in dis- 
tinction from the band, which is a tension-organ. 

17. Modification of Motion. — In the action of a mechanism the 
motion of the follower may differ from that of the driver in kind, in 
velocity, in direction, or in all three. As the paths of motion of the 



6 INTRODUCTION. 

driver and follower depend upon the connections with the frame of the 
machine, the change of motion in kind is fixed, and it only remains for 
us to determine the relations of direction and velocity throughout the 
motion. Now the laws governing the changes in direction and velocity 
can be determined by comparing the movements of the two pieces at 
each instant of their action, and the mode of action will fix the laws. 
Therefore, whatever the nature of the combination, if we can determine 
throughout the motion of the driver and follower, the velocity ratio, and 
directional relation, the analysis will be complete. 

Either the velocity ratio or the directional relation may vary, or 
remain the same throughout the action of the two pieces. 



CHAPTER II. 

COMPOSITION AND RESOLUTION OF VELOCITIES.— MOTIONS OF 
RIGIDLY-CONNECTED POINTS.— INSTANTANEOUS AXIS.— CENTROIDS. 

18. Graphic Representation of Motion. — We can represent the 
motion of a point in any given piece of mechanism, graphically, by a 
right line whose length in units indicates the velocity, and whose direc- 
tion indicates the direction of motion of the point at the instant con- 
sidered; an arrow-head is used to indicate the direction in which the 
point is moving. If the path of the moving point be a curve of any 
kind, the direction of the curve at any point is that of its tangent at that 
point, which indicates the direction of motion as well. 

19. Resultant. — If a material point receives a single impulse in any 
direction, it will move in that direction with a certain velocity. If it 
receives at the same instant two impulses in different directions, it will 
obey both, and move in an intermediate direction with a velocity differ- 
ing from that of either impulse alone. The position of the point at the 
end of the instant is the same as it would have been had the motions, 
due to the impulses, occurred in successive instants. This would also 
be true for more than two motions. The motion which occurs as a con- 
sequence of two or more impulses is called the Resultant, and the sepa- 
rate motions, which the impulses acting singly would have caused, are 
called the Components. 

20. Parallelogram of Motion. — Suppose the point a (Fig. 1) to have 
simultaneously the two component motions represented in magnitude 
and direction by ab and ac. Then the resultant 

is ad, the diagonal of the parallelogram of which 

the component motions ab and ac are the sides. 

Conversely, the motion ad may be resolved into 

two components, one along ab, and the other 

along ac (Fig. 1), by drawing the parallelogram f ig . 1. 

abdc, of which it will be the diagonal. 

Any two component motions can have but one resultant, but a given 

7 




8 



COMPOSITION AND RESOLUTION OF VELOCITIES. 




Fig. 2. 



resultant motion may have an infinite number of pairs of components. 

In the latter case we have a definite solu- 
tion provided we know the direction of both 
components, or the magnitude and direction 
of one. If we know the magnitude of both 
components, there are two possible solutions. 
Thus in Fig. 2, where ad is the given resultant, 
if the two components have the magnitudes 
represented by ac and ab, the directions ac 
and ab would solve the problem, or the 

directions aq and a\ would equally well fulfil the conditions. 
It very often happens that we wish to resolve 

a motion into two components, one of which is 

perpendicular, and the other parallel, to a given 

line, as ef (Fig. 3). Here ad represents the mo- 
tion; ab = ad cos dab, the component parallel to ef; 

and ac = ad sin dab, the component perpendicular 

to ef. 

2i. Parallelepiped of Motions. — If the three component motions 
ab, ac, and ad (Fig. 4) are combined, their resultant af will be the diagonal 

of the parallelopiped of which they 
are the edges. The motions ab and 
ac, being in the same plane, can be 
combined to form the resultant ae; 
in the same way ae and ad can be 
combined, giving the resultant af. 
Conversely the motion af may be 
resolved into the components ab, ac, 
and ad. 

If the parallelopiped is rectangular, the case is simpler, and often 
used; then we have 





Fig. 4. 



af =ae -{-ad =ab -{-ac -{-ad 



To find the resultant of any number of motions : First, combine any 
two of them and find their resultant; then, combine this resultant with 
the third, thus obtaining a new resultant, which can be combined with the 
fourth; and so on. 

22. Velocities of Rigidly-connected Points. — If two points are so 
connected that their distance apart is invariable and if their velocities 
are resolved into components at right angles to and along the straight 
line connecting them, the components along this line of connection must 
be equal, otherwise the distance between the points would change. 

In Fig. 5 let a and ?> be two rigidly-connected points having the l.v. 
of a represented in magnitude and direction by aa, and the l.v. of b in 



VELOCITIES OF RIGIDLY-CONNECTED POINTS. 




direction by bb v The components of aa x perpendicular to and along 

ab are ac and ad respectively. 

The component ad will represent 

the entire tendency of translation 

of the line ab in the direction ab 

due to the l.v. aa x at the point a. 

Since the points a and b are 

rigidly connected, the l.v. of 

any point in the line ab must 

be such that when resolved into 

components perpendicular to and Fig. 5. 

along ab the component along 

ab shall be equal to ad. Therefore the l.v. of b must be bb lf since be must 

be equal to ad. In the figure the motions are shown in one plane, but 

the proposition is also true for motions not in one plane. 

For example, in the series of links shown in Fig. 6, c and d are fixed 
axes and / slides on the line ff x . If aa ± represents the l.v. of a, the com- 
ponent of translation along ab will be am, to which the component bn 
must be equal. Therefore b\ will represent the l.v. of b, where bb x is 





tangent to the path of b in the given position. The l.v. of e will be ee v 
where ee ± is tangent to the path of e in the given position, and where 
ee x \bb 1 = de\db, since in any rotating body the l.v's of any points are 
proportional to their respective distances from the axis. 

To find the l.v. of / we have the l.v. of the point e in ef represented 
by ee x , therefore the component of translation along ef will be eo. The 
component fp must be equal to eo, which gives jf 1 as the resulting l.v. of /. 

23. Instantaneous Axis. — If a line ao (Fig. 5) is drawn through a 
perpendicular to the direction of motion aa t of the point a, then the 
motion of a may be the result of a rotation about an axis through any 
point in the line ao or in ao produced. Similarly, the motion bb t may be 
the result of a rotation about an axis through any point in bo. If a and b 
are rigidly connected, the piece on which they are situated must have a 



10 COMPOSITION AND RESOLUTION OF VELOCITIES. 

rotation about one axis, and the a.v. of all points about that axis must 
be the same. The only point satisfying this condition is o, at the inter- 
section of ao and bo, and the piece ab has a motion at that instant »such 
as it would have if it were rotating about an axis through o. The axis 
through o, perpendicular to the plane of the motions, is called the instan- 
taneous axis, it being the axis about which the body is rotating for the 
instant in question. 

The a.v. about the instantaneous axis being the same for the instant, 
for the points a and b, the l.v's of a and b will be proportional to their 
distances from the instantaneous axis; 

.'. aa 1 :bb 1 = oa:ob. 

If the motions of the points a and b are not in the same plane, the 
instantaneous axis would be found as follows: Pass a plane through 
the point a perpendicular to aa t ; the motion aa x might then be the result 
of a revolution of a about any axis in that plane. In the same manner, 
the motion of bb x might be the result of a revolution of b about any axis 
in the perpendicular plane through b. The points a and b, being rigidly 
connected, must rotate about one axis, which in this case will be the 
intersection of the two perpendicular planes. 

Suppose the motions of the two points a and b to be in the same 
& - plane and parallel, as in Figs. 7 and 8. Here 

the perpendiculars through a and b coincide 
and the above method fails. Let aa x and 

~~^Z 1 b\ be the l.v's of the points a and b respect- 

jf'^ ively. To find the instantaneous axis draw 

a right line through the points a ± and b x in 
each case and note the point o where it inter- 
&! sects ab or ab produced. This must be the 

instantaneous axis, for from the similar tri- 
angles aa x o and b\o we have 






::l 



a ?• aa.:bb.=oa:ob, 

Fig. 8. , * /•-,,, 

the same equation as was obtained before. 

In the solid, illustrated by Fig. 9, given the l.v. of a in magnitude 
and direction aa 1} also the direction of the l.v. of b, bb v to find the l.v. 
of the point c. Knowing the directions of the motions of a and b at the 
given moment we find the instantaneous centre at o ; therefore the direc- 
tion of the motion of c must be cc v To find the magnitude of the l.v. of 
c we have 

cc l :aa 1 = co:ao. 

Or we may determine cc x by finding the component of aa ± along ac, which 
will be ae, and the component of cq along ac must be the same or cf. 
There is still another method of solution, not using the instantaneous 
centre. We find that the l.v. of c must have a component cf along ac, 



INSTANTANEOUS AXIS OF ROLLING BODIES. 



11 



its other component being perpendicular to ac; but after determining 
the l.v. of b, bb 1} we find that we must also be able to resolve the l.v. 
of c into rectangular components, one of which, ck, shall be along be, and 




Fig. 9. 

equal to bh, the component of b\ along be. Drawing perpendiculars 
from / and k to ac and be respectively, their intersection c x will give the 
l.v. of c, cc v which will answer the above requirements. 

24. Instantaneous Axis of Rolling Bodies. — The instantaneous axis 
of a rigid body which rolls without slipping upon the surface of another 
rigid body must pass through all the points in which the two bodies 
touch each other; for the points in the rolling body which touch the 
fixed body at any given instant must be at rest for the instant, and 
must, therefore, be in the instantaneous axis. As 
the instantaneous axis is a straight line, it follows 
that rolling surfaces which touch each other in 
more than one point must have all their points of 
contact in the same straight line in order that no 
slipping may occur' between them. This property 
is possessed by plane, cylindrical, and conical 
surfaces only; the terms cylindrical and conical 

being used in a general sense, the bases of the cylinders and cones having 
any figure as well as circles. 

Let Fig. 10 represent a section of the rolling surfaces by a plane per- 
pendicular to their straight line of contact, and assume pp as fixed; 
then is a point in the instantaneous axis, as it is for the instant at 
rest, and all points on C, as a and b, are rotating about it for the instant. 
To find the direction of motion of any point, as a, draw ao, and through 
a perpendicular to ao draw aa v which is the direction of motion of a 




A 



12 COMPOSITION AND RESOLUTION OF VELOCITIES. 

for the instant. The linear velocities of a and b are proportional to their 
distances from o, the instantaneous axis. 

25. Motion of Translation. — If, in Fig. 8, the two parallel motions 
aa t and bb 1 become equal to each other, then ob will be infinite and the 
consecutive positions of ab will be parallel to each other. This is also 
true if the motions are at any angle with ab, so long as they are equal 
and parallel, as in Fig. 11. 

The motion of a line, or of a body containing that line, at any instant 

when it is thus revolving about an axis 
& L at an infinite distance, is called translation. 
All points in such a body move in the 
same direction with the same velocity; 
Fig. 11. tne paths of the points may be rectilinear 

or curvilinear. Straight or rectilinear 
translation is commonly called sliding. As an example of straight trans- 
lation, we have the cross-head of a steam engine; of curvilinear transla- 
tion, the parallel-rod of a locomotive. 

26. Periodic Centre of Motion. — It very often happens that we 
know two positions of a line, as ab and afi^ (Fig. 12) 3 moving in the 
plane of the paper, and we wish to find an , h 
axis about which this line could revolve to 
occupy the two given positions. Draw 
aa x and bb lf and find the intersection of the 
perpendiculars drawn at their middle points. \ i j ^>&i 
Thus ab can be brought to the position a 1 6 l \ 1/ / /^ 
by revolving it about an axis through \ \ // 'Z 
perpendicular to the plane of the paper, 6y^ 

the paths of a and b being arcs of circles /] 

drawn from as a centre, and with radii j _ 

equal to oa and ob respectively. 

When the two positions of ab are taken infinitely near each other, 
becomes the instantaneous centre. 

27. Centroid. — The curve passing through the successive positions 
of the instantaneous centre of a body having a combined motion of 
rotation and translation is called a centroid. The surface formed by the 
successive positions of the instantaneous axis is called an axoid. 

Suppose we know the relative motions of two links as ab and cd in 
the mechanism given in Fig. 13, where the motion of ab relative to cd, 
cd being considered as fixed, is such that a moves in the path a 3 aa 2 , 
and b moves in the path b 3 bb 2 , a 3 b 3 , a 2 b 2 , etc., being positions of ab. If 
in any of these positions, as ab, we draw from a and b normals to their 
respective paths, their intersection will be the instantaneous centre 
of ab for that position. A smooth curve passed through the successive 



CENT ROW. 



13 



positions of the instantaneous centre, o, o,, o„ etc., will be the centroid 
of ab. In § 24 we saw that the 
instantaneous axis of one body 
rolling on another was at their 
point of contact. From this 
it would follow that, consider- 
ing one body as fixed relative 
to the other, its surface would 
be the axoid of the moving 
body. Therefore, in Fig. 13, 
the axoid of ab, which is repre- 
sented by the centroid o x oo 2 , 
may be taken as the surface of 
a fixed body, containing dc, on 
which the surface of a moving 
body, containing ab, shall be 
able to roll, giving the same 
motion to ab as the original 
links would give. 

To find the trace of the sur- 
face of the body containing ab, 
we have in each of the positions which it may occupy, distances from a and 
b to its instantaneous centre for that position, which distances are, there- 
fore, distances from a and b to a point in the trace of the surface of the 
body containing ab. Thus am 1 and bm 1 are equal respectively to a l o l and 
b^; similarly am 2 and bm 2 are equal respectively to a 2 o 2 and b 2 o 2 . A 
smooth curve through these points om{m 2 , etc., would give the trace of the 
surface of the body containing ab. It will also be found that this curve 
om x m 2 is the centroid of cd relative to ab, when ab is assumed fixed. 




Fig. 13. 




CHAPTER III. 

PAIRS OF ELEMENTS.— BEARINGS AND SCREWS.— WORM AND WHEEL. 

28. Pairs of Elements. — In order that a moving body, as A (Fig. 14) 
may remain continually in contact with another body B, and at the same 

time move in a definite path, B would 
have a shape which could be found by 
allowing A to occupy a series of consecu- 
tive positions relative to B, and drawing 
the envelope of all these positions. Thus, 
if A were a parallelopiped, the figure of 
FlG - 14 - B would be that of a curved channel. 

Therefore, in order to compel a body to move in a definite path, it must 
be paired with another, the shape of which is determined by the nature 
of the relative motion of the two bodies. 

A machine consists of elements which are thus connected in pairs, 
the stationary element preventing every motion of the movable one 
except the single one desired. 

Closed Pair. — If one element not only forms the envelope of the 
other, but encloses it, the forms of the elements being geometrically 
identical, the one being solid or full, and the other being hollow or open, 
we have what may be called a closed pair. The pair represented in Fig. 
14 is not closed, as the elementary bodies A and B do not enclose each 
other in the above sense. 

On the surfaces of two bodies forming a closed pair we may imagine 
coincident lines to be drawn, one on each surface; and if we suppose 
these lines to be such in form as will allow them to move along each 
other, that is, allow a certain motion of the two bodies paired, we shall 
find that only three forms can exist: 

1° A straight line, which allows straight translation. 

2° Among plane curves, or curves of two dimensions, a circle, which 

allows rotation. 
3° Among curves of three dimensions, the helix, which allows a 
combination of rotation and straight translation. 

14 



PRIMARY AND SECONDARY PIECES. 15 

29. Primary and Secondary Pieces. — In order to distinguish between 
pieces of a machine which are connected directly to the frame and those 
carried by other moving pieces, the former are called primary, and the 
latter secondary pieces. 

Thus, if the connection of the primary pieces to the frame be by 
closed pairs of elements, the following determinate motions can be 
given to them: 

1° Straight translation or sliding; 
2° Rotation, motion in a circle, as a wheel on its axis; 
3° A helical motion, which might be considered as a combination of 
1° and 2°, as a screw. 

30. Bearings are the surfaces of contact between the frame and the 
primary pieces, the name being applied to the surface of each piece; but 
these surfaces sometimes have distinctive names of their own. 

The bearings of primary pieces may be arranged, according to the 
motions they will allow, in three classes : 

1° For straight translation the bearings must have plane or cylin- 
drical surfaces, cylindrical being understood in its most general sense. 
The surfaces of the moving pieces are called slides; those of the fixed 
piecei, slides or guides. 

2° For rotation, or turning, they must have surfaces of revolution, as 
circular cylinders, cones, conoids, or flat disks. The surface of the solid 
or full piece is called a journal, neck, spindle, or pivot; that of the 
hollow or open piece, a journal, gudgeon, pedestal, plumber- or pillow- 
block, bush, or step. 

3° For translation and rotation combined, or helical motion, they 
must have a helical or screw shape. Here the full piece is called a screw, 
and the open piece a nut. 

It will be interesting to note the relation that the slide and journal 
bear to the screw, from which they might be considered as derived. 
If we suppose the pitch of a screw /v 

to be diminished until it becomes 




zero, or if we suppose the pitch 

angle to become zero, then the form 

A (Fig. 15) would be changed to that 

of B, which, with a modification of 

the thread outline, would become, like C, a common form for a journal. 

Thus, by making the pitch zero, the motion along the axis of the screw 

has been suppressed, and only rotation is possible for the nut. If we 

suppose the pitch angle to increase instead of diminish, the screw will 

become steeper and steeper. If the angle =90°, the screw-threads 

become parallel to the axis, the screw becomes a prism, and the nut a 

corresponding hollow prism, as Fig. 15, D. Here rotation is suppressed, 



16 



BEARINGS AND SCREWS.— WORM AND WHEEL. 



and only sliding along the axis is possible, giving us the slide. If the 
angle be made >90°, the screw changes from a right- to a left-handed 
one, but still remains a screw. 

It is very often the case that pulleys or wheels are to turn freely on 
their cylindrical shafts and at the same time have no motion along them ; 
for this purpose, rings or collars (Fig. 16, A) are used, the collars D and 
E, held by set screws, prevent the motion of the pulley along the shaft 
but allow its free rotation. Sometimes pulleys or couplings must be free 

A M 



c 



e^ 




m 

Fig. 16. 
to "slide along their shafts, but at the same time must turn with them; 
they must then be changed to a sliding pair. This is often done by 
fitting to the shaft and pulley or sliding piece a key C (Fig. 16, B), 
parallel to the axis of the shaft. The key may be made fast to either 
piece, the other having a groove in which it can freely slide. The above 
arrangement is very common, and is called a feather and groove or spline, 
or a key and key way. 

31. Screw and Nut. — A screw might be defined as a solid cylindrical 
body with a thread or projection of uniform section wound around it in 
successive equidistant coils or helices ; a nut would be formed b}^ winding 
the thread on the inside of a hollow cylinder. Either the screw or the 
nut may be the moving piece, the nut being the envelope of the screw 
in all cases. 

The form of the section of a screw-thread varies with the use to 
which the screw is to be put; Fig. 17. shows some of the common forms. 




Fig. 17. 

The most common form is shown at A, and is known as the V thread, 
its section being an equilateral triangle. As the sharp edges make the 



SCREW-PITCH. 17 

thread liable to injury, and less easy to construct, the modified forms 
B and C are much used. 

Form B, known as the Sellers or United States standard, has the 
angle of the thread 60°, and one-eighth of the depth of the V cut off 
at the top and at the bottom; this makes a better screw, as more material 
is left between the bottoms of the threads, the very thin parts removed 
being of little use as bearing surfaces on account of their weakness. 
Form C, known as the Whit worth, or English standard, has the angle 
of the thread 55°, and one-sixth of the depth of the V is rounded off 
at the top and at the bottom of the thread. 

As the resistance of pieces in sliding contact is normal to the bearing 
surfaces, there is a tendency in all V-shaped threads to burst the nut. 

D shows the square-threaded screw, most commonly used to produce 
motion, as it has large wearing surfaces perpendicular to the motion 
given by the screw; it is, however, not so strong as A, as it has only 
one-half the shearing surface of A in a given length of the nut. E is a 
combination of B and D, used for screw gearing and the lead screws <4i 
engine-lathes. This thread with an angle of 29° is now known as the 
Acme standard. With ordinary pitches this angle will permit a clasp- 
nut to be used on the screw as in engine-lathes, which is not possible 
with D. A modified form of D, used for rough work, the screws being 
cast, is shown at F. In G, which is used w T here the force is always 
applied in the same direction, as in the breech-screws of large guns, 
the shearing strength of A is combined with the flat bearing surface 
of D. Lag-screws, used in wood, have the form of thread shown at H; 
here the wood is the weaker material and has the larger thread. 

Right- and Left-handed Screws. — A screw is said to be right-handed 
or left-handed according as a right-handed or left-handed rotation is 
required to make it advance; and this is a permanent dis- 
tinction. In Fig. 18, R shows a right-handed and L a 
left-handed screw; it will be noticed in R that when a 
right-handed screw is held vertically, the threads will rise 
from the left to the right on the visible side; in the left- 
handed screw the reverse is the case. 

32. Screw-pitch. — The pitch of a screw is the distance it advances 
for one complete turn, and, in single-threaded screws, is measured by 
the distance between two similar points on successive threads measured 
on a line parallel to the axis of the screw. Such screws are commonly 
designated by the number of threads to the inch of length; that is, a 
screw of T y pitch, or ten threads to the inch, is called a screw of ten 
threads per inch. 

33. Multiple-threaded Screws. — If, instead of winding one thread 
around a cylinder, several equidistant threads are wound at the same 




18 



BEARINGS AND SCREWS.— WORM AND WHEEL. 



time, taking care in the winding that the threads are kept the same dis- 
tance apart, we shall have a multiple-threaded screw. If two threads are 
used, a double-, and if three threads, a triple-threaded, screw will result, 
and so on. By the above principle, the pitch can be greatly increased 
without necessarily increasing the size of the thread. Here the pitch is 
measured by the axial distance between two similar points on successive 
coils of the same thread, one point being found from the other by follow- 
ing the thread for one complete turn. 

34. Velocity Ratio. — A screw may be used to produce motion in two 
ways: 

1° The nut may be fixed, and the screw be made to turn by apply- 
ing a force at the end of a lever, or on the circumference of a wheel 
attached to the screw. While the screw advances through a distance 
equal to the pitch, the point at which the force is applied describes one 
coil of a helix of equal pitch. If P represents the pitch of the screw, 
and R the shortest distance between the point of application of the force 
and the axis of the screw, called the lever-arm, the velocity ratio is 

P 

2° The screw may simply rotate, and the nut may have a motion of 
translation in a straight line without turning. While the screw makes 
- w one turn, the nut will move through a distance equal 

to the pitch, and the point of application of the 
force will describe a circle of radius R; the veloc- 
ity ratio is 

2nR 
P ' 

The latter form for the velocity ratio is, on ac- 
count of its simplicity, used as an approximation to 
the first. 

Either of the above combinations may be re- 
versed, that is, the nut may be made to turn and 
the screw remain stationary in 1°, or have a straight 
translation in 2°. This does not change the veloc- 
ity ratio. 
For example, in the case of a simple jack-screw as in Fig. 19, if 
P is the pitch of the screw, and R the length of the lever- arm, we 
have 



s 



E5U 



Fig. 19. 



l.v. of F = V(2tiR 2 +P 2 ) ^27:R 
l.v. of W P P 



nearly. 



(5) 



35. Compound or Differential Screws.— If A (Fig. 20) is a fixed 
nut carrying the screw S, and B is a movable nut, also on the screw 



COMPOUND OR DIFFERENTIAL SCREWS. 



19 



S, and free to slide along the guides GG, the pitches of the screw in A 
and B being P x and P 2 respectively, 



0Jig 




Fig. 20. 



ifiii sii 



Fig. 21. 



P 2 being smaller than P x and both 
threads being right-handed; we shall 
have for each turn of the screw in the 
direction of the arrow an advance of 
the screw S to the right equal to the 
pitch P v Meanwhile the nut B has 
moved relatively to the screw a dis- 
tance P 2 to the left. The absolute 
motion of B is then to the right and 
equal to (P 1 — P 2 ), the resultant of 
its motion relatively to S, and the 

motion of S. The same result would be obtained by supposing the 
nuts A and B to act in succession. Thus, suppose B fast to the screw 
and free to turn, then one turn of the screw in A would advance B a 
distance +P t (motion to the right being positive); now suppose the 
screw fast in A, and turn the nut B back one turn to the position it would 
have had provided it had not rotated; B will then move a distance 
— P 2 . Adding the two motions, we have for the motion of B, (P 1 — P 2 ) 
as before. This principle of successive movements is very often con- 
venient in determining resultant motions. 

When the resultant motion is, as above, the difference of two com- 
ponent motions, the screw is called a differential screw. 

If one of the threads on S is right-handed and the other left-handed, 
the motion of B would be (P 1 +P 2 ), its direction depending on the 
arrangement and rotation of the screw. A right- and left-handed screw 
are often used in combination to bring together two pieces, not capable 
of turning, as in the right and left pipe-coupling. Pieces can also be 
arranged so as to move equal distances in opposite directions in reference 
to some point located between them. 

A more practical form of differential screw than Fig. 20 is shown in 
Fig. 21, where the screw S v working in the fixed nut A, is made hollow, 
and forms the nut, for the smaller screw, S 2 , which is fast to the slide B, 
moving on the guides GG. The action is the same as in the previous 
case. 

In all the previous cases the force has been applied to rotate the 
screw or nut, and thus cause a straight translation ; a force causing trans- 
lation might be applied to the screw or nut, which would cause the nut 
or screw to rotate. This is not possible with ordinary pitches, as the 
frictional resistance is so great; it is well known, however, that nuts and 
screws subjected to constant jarring, such as those on railway trucks, 
are very liable to work loose; and double nuts, one serving as a check 
for the other, are often used. When the pitch is made very long, the 



20 BEARINGS AND SCREWS.— WORM AND WHEEL. 

screw can be easily turned by moving the nut along it; in this case the 
screw is formed by a steep spiral groove running along a cylindrical 
piece. The nut fits this cylindrical piece, and has a projecting feather 
which fits the groove. This principle is used in a small automatic drill, 
where the spindle which carries the drill has a multiple- threaded screw 
of rapid pitch, cut about two-thirds of its length. This screw fits into 
a tubular handle closed at one end and furnished with a nut which 
fits the screw: by pushing upon the handle, the screw with the drill 
is made to rotate; a coiled spring placed between the end of the screw 
and the closed end of the tube returns the screw to its normal position. 

36. Screws are correctly cut in a lathe where the cylindrical blank 
is made to rotate uniformly on its axis, while a tool, having the same 
contour as the space between the threads, is made to move uniformly on 
guides in a path parallel to the axis of the screw, an amount equal to 
the pitch for each rotation of the blank. The screw is completed by 
successive cuts, the tool being advanced nearer the axis for each cut 
until the proper size is obtained. A nut can be cut in the same way by 
using a tool of the proper shape and moving it away from the axis for 
successive cuts. 

Screws are also cut with solid dies either by hand or power, and with 
proper dies and care good work will result. Nuts are generally threaded 
by means of "taps" which are made of cylindrical pieces of steel having 
a screw-thread cut upon them of the requisite pitch; grooves or flutes are 
then made parallel to the axis to furnish cutting edges, the tap is then 
tapered off at the end to allow it to enter the nut, and the threads are 
"backed off" to supply the necessary clearance. 

Screws cut by open dies that are gradually closed in as the screw is 
being cut are not accurate, as the screw is begun on the outside of the 
cylinder by the part of the die which must eventually cut the bottom of 
the thread on a considerably smaller cylinder. Thus, as the angle of tne 
helix is greater the smaller the cylinder, the pitch remaining the same, 
the die at first traces a groove having a pitch due to the greater angle 
of the helix at the bottom of the thread. As the die-plates are made to 
approach each other, they tend to bring back this helical groove to the 
standard pitch; this strains the material of the threads, and finally pro- 
duces a screw of a different pitch than that of the die-plates. 

37. Worm and Wheel.— A worm and wheel (Fig. 22) is a combina- 
tion of a screw and a wheel furnished with teeth so shaped as to be 
capable of engaging with the screw placed tangential to the wheel. The 
continuous rotation of the screw or worm will then impart continuous 
rotation to the wheel, and it will advance through one, two, or three 
teeth upon each turn of the screw, according as the thread 
on the screw is single, double, or triple. On account of the great 



POWER OF A SCREW. 



21 



reduction of velocity obtainable by this combination, it is ex- 
tremely valuable as a means of obtaining 
mechanical advantage, and is much used 
in hoisting machinery. It is also useful 
in making fine angular adjustments, as 
in gear-cutting machines; when thus used 
for making adjustments, it is sometimes 
called a " tangent-screw. " 

Velocity Ratio in a Worm and Wheel. — 
Two cases may be considered : 

1° Let P be the pitch of the worm, D x 
the pitch diameter of the worm-wheel, D 2 
the pitch diameter of the drum upon which 
the resistance W is exerted . For one turn of 
the worm the point where the force F is 
applied will move a distance 2tzR and the surface of the drum where 

■W is exerted will move PX- 




Fig. 22. 



2 

d; 



l. v. of F 
l.v. of W 



motion of F 
motion of W 



2nR 
IX 



(6) 



2° Let the worm be double-threaded and let N be the number of 
teeth on the worm-wheel, then we shall have 

l.v. of F 2tzR 



l.v. of W 



2 
N 2 



(7) 



The denominator -^tzD 9 would be the motion of W for one turn of the 

N 2 

worm. If the worm were triple-threaded, the motion of W for one turn 

3 
of the worm would be ^tzD 2 . 

38. Power of a Screw. Relation between Forces and Linear Veloci- 
ties in Mechanisms. — Since, if we neglect the loss of work by friction or 
concussion, any mechanical combination must deliver as much work as 
> it receives, we must have the force at the point of application multiplied 
by the velocity of that point equal to the force at the point of delivery 
multiplied by its velocity, or the forces are to each other inversely as 
the velocities of the points at which they act. For example, in the 
previous paragraph we have 

l.v.F _W 

UW~Y (S) 

Combining this fact with the previous statement in regard to the ratio 
of the linear velocities of the points where the forces act, we can find 



22 BEARINGS AND SCREWS.— WORM AND WHEEL. 

the forces which may be transmitted in such cases, when losses due to 
friction are neglected. Thus in case 1° in the preceding paragraph 
we should have 

W l.v.F 2tzR 

F ~l.v. W~ Ty D 2 (9 > 

39. Inversion of Closed Pairs. — If, in a closed pair, we exchange 
the fixed element for the movable one, there is no alteration in the 
resulting absolute motion; the exchange of the fixedness of an element 
with its partner is called the inversion of the pair. This has already 
been noticed in connection with the discussion of the screw, where it 
made no difference in the resulting motion whether the screw or the nut 
was considered as fixed. In the ordinary bolt we turn the nut, while in 
the " tap-bolt" the nut is stationary and the bolt is turned. In the 
common wagon- wheel the axle is fixed to the body of the wagon, while 
the wheel turns on it ; in the railway truck the bearing is attached to the 
truck frame, and the axle turns in it with the wheel, which is made fast 
to the axle. 

40. Incomplete Pairs of Elements ; Force closure. — Hitherto it has 
been assumed that the reciprocal restraint of two elements forming a 
pair was complete, i.e., that each of the two bodies, by the rigidity of 
its material and the form given to it, restrained the other. In certain 
cases it is only necessary to prevent forces having a certain definite 
direction from affecting the pair, and in such cases it is no longer abso- 
lutely necessary to make the pair self -closed ; one element can then be 
cut away where it is not needed to resist the forces. When a pair of 
elements is thus incomplete, and the closure is effected by means of a 
force or forces, we have what is called a force-closed pair of elements. 

The bearings for railway axles, the steps for water-wheel shafts, the 
ways of an iron planer, railway wheels kept in contact with the rails by 
the force of gravity, are all examples of force-closed pairs. 



CHAPTER IV. 

ROLLING CYLINDERS AND CONES CONNECTED BY FORCE-CLOSURE.— 
ROLLING OF NON-CYLINDRICAL SURFACES.— LOBED WHEELS. 

41. One of the most common and most useful problems of mechan- 
ism is to connect two axes so that they shall have a definite angular 
velocity ratio. We may have three cases: 1° parallel axes; 2° axes 
which intersect; 3° axes which are neither parallel nor intersecting. In 
any case the velocity ratio may be constant or it may vary. The first 
two cases, with a constant velocity ratio and directional relation, are the 
most common, and resemble each other in many ways; for that reason 
they will be considered first; the third case will come up later. 

Pure Rolling Contact consists of such a relative motion of two lines 
or surfaces that the consecutive points or elements of one come suc- 
cessively into contact with those of the other in their order. 

42. Rolling Cylinders. — Let A and B (Fig. 23) be the sections of 
two rolling cylinders, and let o x and o 2 , the 
centres of the circles A and B, be the traces 
of their axes. The point c is called the 
point of contact. The common element 
through c is called the line or element of 
contact. The plane passing through the 
axes, whose trace is the common normal F 

through the point c. is called the plane of 

centres; the common normal o x o 2 is called the line of centres. 

According to the definition of rolling contact, the linear velocity 
ratio of the two rolling surfaces must be unity, or slipping will result 
between them. 

43. Angular Velocity Ratio. — In Fig. 23 let R 1 and R 2 be the radii of 
two rolling cylinders, V 1 and V 2 their angular velocities, and N x and N 2 
their revolutions per minute. Suppose the two circles to roll in such a 
way that the point c travels to d in A, and to e in B; then, as no slipping 

arc cd = arc ce, or R ± V x = R 2 V 2 . 

••• £-! •••••••• aw 

23 




24 



ROLLING CYLINDERS AND CONES, ETC. 



That is, when two circles roll together, their uniform angular velocities 
are inversely as the radii of the circles. 

Now if, as is more commonly the case, we take the number of revo- 
lutions per minute, N t and N 2 , as given, the l.v's per minute of the 
rolling surfaces would be 2t:R 1 N 1 and 2nR 2 N 2 , which must be equal. 



2nR 1 N l = 2<n:R 2 N 2 , or RJN^Rfl^ 

N x R 2 



N 2 Ri 



(11) 



That is, when two circles roll together, their revolutions in a given time 
are inversely proportional to the radii of the circles. 

44. Given the velocity ratio and the distance between centres of a 
pair of rolling cylinders, to find their radii. 

External Contact (Fig. 23). — If D is the distance between the axes, and 
A makes N x revolutions and B makes N 2 revolutions per minute, we have 
from equation (11) 

R,^ 1 



and , from the figure, R^ +R 2 = D. 

Solving the two equations for R x and R 2 , we have 




*,- 



N 2 D 

N 2 +N, 



and R = 



n 2 +n; 

Internal Contact (Fig. 24). — Using the same 
notation as above, we have 



R 2 



El 
N, 



and 



R, 



R 2 =D. 



Fig. 24. 



Ri- 



NJ) 



and R« 



N t D 



Graphical Solution. — External Contact. — Let o x o 2 be the line of centres 
(Fig. 25). Through o ± draw the line o x n, and lay off upon it, to some 
convenient scale, the distances o x m and 
mn equal to the number of units in N 2 
and N 1 respectively. Draw no 2 through 
the centre o 2 , and then draw mc parallel 
to no 2 . c will be the point of contact 
and R t and R 2 the radii. 

Internal Contact (Fig. 26). —Draw the line 
0{n as before, and lay off o x ra equal to the 
units in iV 2 ; 




Fig. 25. 
then lay off mn toward o t from m equal to the units in N x ; 



RACK AND FIN ION. 



25 



/ 
| 4 Ri- 


$ 


*\l °t 


™-Bri 


>». n/ 


//' 




^-'/ / 




^V / 




VI / 




/ 



Fig. 26. 



draw no 2; and mc parallel to no 2 ; R t and i? 2 will be the radii,, and c 
the point of contact. \ 

It will be noticed that two circles in exter- ^ .- — ^ \ 

nal contact rotate in opposite directions, 
while those in internal contact rotate in 
the same direction. 

The various forms of roller bearings, 
where a series of rolling cylinders occupy the 
annular space between two coaxial bear- 
ing cylinders and roll externally on one and 
internally on the other are good examples of rolling cylinders. 

We shall find in the study of gearing that it is possible to use teeth 
developed from rolling cylinders, which teeth may be of such shape 
that the conditions of pure rolling contact of the cylinders shall be prac- 
tically obtained. 

45. Rack and Pinion. — If we suppose the circle B (Fig. 23) to be 
indefinitely increased, its radius will eventually become infinite, and its 
circumference will become sensibly a straight line. This combination is 
often used in gearing, and the velocity ratio is properly considered 
here. As the straight line or rack has straight translation only, its 
linear motion needs only to be considered. If the radius R, and the 
a.v. or number of revolutions N, of the rolling circle or pinion are given, 
the l.v. of the rack L is found by the equation 

L = 2-RN. 
For, to satisfy the conditions of rolling contact, the l.v's must be 
equal. Here the velocity ratio is constant, but the directional relation 
must be reversed when the end of the rack reaches the pinion. 

46. Rolling Cones. — Let A and B (Fig. 27) be a pair of rolling cones ; 

ce, the element of contact; o x co 2 e, the 
plane of centres. Let o x cs and o 2 ct be 
planes passing through the point of 
contact c and respectively perpen- 
dicular to o x e and o 2 e; the circles cut 
from the cones by these planes are 
called base circles. Similar circles 
could be cut by planes parallel to 
o^cs and o 2 ct through any other point 
on the element of contact, as d '. 
Now in order to have no slipping, the 
linear velocity ratio of all pairs of 
points on the common element of con- 
tact ce must be unity, that is, the 
consecutive elements of one cone must 

roll successively into contact with these of the other. From this it 




Fig. 27. 



26 



ROLLING CYLINDERS AND CONES, ETC. 



also follows that two or more cones in pure rolling contact must have a 
common apex. 

47. Angular Velocity Ratio. — Let R t and R 2 be the radii of the base 
circles, N x and N 2 be their revolutions per minute, and V 1 and V 2 their 
angular velocities. Since the l.v's of the two base circles in contact at 
c are the same, we have, as for rolling circles, 

R 2 V, AY 
and for any other point, as c', we must also have 

R^JJ^J^ 

R 2 ' V t Nj 
as R x and R t ', being in one piece, must revolve with the same a.v. about 
o x e. ■ It can also easily be seen by similar triangles that 

Ri = Rl 
R 2 R 2 ' 

Thus the angular velocity ratio of any two circles in contact on the 
common element is the same as that of the base circles and that of the 
two axes. 

In practice it is customary to use thin frusta of the rolling cones for 
rolling conical wheels, and we shall find in this case also that teeth can 
be developed from the cones and give the same result as pure rolling 
contact of the original cones. Such toothed wheels are called bevel gears. 

Given the a.v. ratio, the position of the base of one of the cones, 
and the angle between the axes: to find the position of the base of the 
other cone, and the radii of both bases. 

Algebraical Solution. — Let o x e and o 2 e (Fig. 28) be the axes of a pair 
of rolling cones, 6 the angle between the axes, R^ and R 2 the radii of 

bases, N 1 and N 2 the revolutions per 
minute, and a x and a 2 the semi- 
vertical angles of the respective cones. 

R x = ec sin a^ ; 

R 9 = ec sin a 2 ; 

R 2 AY 




\j 2 

sin a x 
sin a 2 



(12) 



sm a: j 
cos a, 



substituting for a 2 the value (6— a x ) 
and solving, we have 

N 2 _ sin a x _ sin a t 

N t sin(# — oY sin# cosa^ — cos sin c^ 

tan a t 



sin 6 — cos 



sin «! sin 6 — cos 6 tan a x ' 
cos a 1 



ANGULAR VELOCITY RATIO. 



27 



.*. tan a t = 



N 2 sin 



sin# 



N x +N 2 cos N lt a 
^+cos# 

In a similar manner we could find for a 2 the equation 

sin# 



tan a, = 



*3 



(13) 



(14) 



+ COS0 



Now the two angles a t and a 2 being known, and the distance o x e of one 
base circle from the intersection of the axes being given, the distance 
o 2 e of the other base circle from the intersection of the axes may be 
readily found and also the radii R x and R 2 of the base circles. 

The angle between the axes is often 90°, in which case we should 
have 

tan «i = ^-; tana 2 =-^; 

N 
R x = o x e tan a x = o^e^- ; R 2 = o x e. 

Graphical Solution. — Given the angle between the axes of two roll- 
ing cones and their a. v. ratio: to find the element of contact of the 
two cones. Let N t and iV" 2 be the revolutions per minute of the axes 
o x e and o 2 c respectively (Fig. 29). Then, since the revolutions are 



.--1 



V^^ ^ 


a 


o 9 


*xT"\X 

\ x — ^ 


b 




\ X ♦ 


/V 




c 




Fig. 29. 

inversely as the radii at any point of contact, the element of contact 
ec may be found by erecting, to any convenient scale, perpendiculars 
to o t e and o 2 e inversely as their respective revolutions. Thus ab repre- 
sents JV, and dg represents N 2 . The intersection / of the lines drawn 
-through the ends of these perpendiculars parallel to the respective 



28 



ROLLING CYLINDERS AND CONES, ETC. 



axes will give one point on the element of contact, which may then 
be drawn through the point thus found, and through e. 

Fig. 30 shows a case in which the data are such that the cones are 
found to be in internal contact. 

48. Rolling Cylinder and Sphere.— Fig. 31 shows an example of a 
rolling cylinder and sphere as used in the Coradi planimeter. The seg- 
ment of the sphere A turns on an axis ac passing through a, the centre 
of the sphere. The cylinder B, whose axis is located in a plane also 




Fig. 31. 

passing through the centre of the sphere, is supported by a frame pivoted 
at e and is held to the cylinder by a spring, not shown. The frame 
pivots e are movable about an axis at right angles to ac and passing 
through a, the centre of the sphere. When the roller is in the position 
B with its axis at right angles to ac, the turning of the sphere produces 
no motion of B; when, however, the roller is swung so that its axis 
makes an angle bac 1 with its former position, as shown at B x by fine 
lines, the point of contact is transferred to c x in the perpendicular from 
a to the roller axis. If now we assume the radius of the roller =R, the 
relative motion of roller and sphere, in contact at c x , is the same as that 
of two circles of radii R and bc ± respectively. Transferring the point of 
contact to the opposite side of ab will result in changing the directional 
relation of the motion. The action of this device is purely rolling and 
but very little force can be transmitted. It is used only in very delicate 
mechanisms. 

Disc and Roller. — If in Fig. 31 we assume the radius of the sphere 
ac to become infinite and the roller B to be replaced by a sphere of the 
same diameter turning on its axis, we shall have a disc and roller as 
shown in Fig. 32, where A A represents the disc and B the roller, made 
up of the central portion of the sphere. 

If we suppose the rotation of the disc to be uniform, the velocity 
ratio between B and A will constantly decrease as the roller B is 






FRIC TION-GEA RING. 



29 



= 

v R, a 



Fig. 32. 



shifted nearer the axis of A, and conversely. If the roller is carried 
to the other side of the axis, it will rotate 
in the opposite direction to the first. c 

This combination is sometimes used 
in feed mechanisms for machine tools, c 
where it enables the feed to be adjusted 
and also reversed by simply adjusting the 
roller on the shaft CC. 

49. Friction-gearing. Rolling cylin- 
ders and cones are frequently used to 

transmit force, and constitute what is known as friction-gearing. In 
such cases the axes are arranged so that they can be pressed together 
with considerable force, and, in order to prevent slipping, the surfaces 
of contact are made of slightly yielding materials, such as wood, leather r 
rubber, or paper, which, by their yielding, transform the line of contact 
into a surface of contact, and also compensate for any slight irregulari- 
ties in the rolling surfaces. Frequently only one surface is made yield- 
ing, the other being usually made of iron. As slipping is likely to take 
place in these combinations, the velocity ratio cannot be depended upon 
as absolute. 

When rolling cylinders or cones are used to change sliding to rolling 
friction, that is, to reduce friction, their surfaces should be made as hard 
and smooth as possible. This is the case in roller bearings previously 
described and in the various forms of ball bearings where spheres are 
arranged to roll in suitably constructed races, all bearing surfaces being 
made of hardened steel and ground. 

Friction-gearing is utilized in several forms of speed-controlling 
devices, among which the following are good examples: 

Fig. 33 shows the mechanism of the Evans friction-cones, consisting 

of two equal cones A and B turning 
on parallel axes with an endless 
movable leather belt, C, in the 
form of a ring running between them, 
the axis of B being urged toward A 
by means of springs or otherwise. 
By adjusting the belt along the 
cones, as shown by the arrows, their 
a.v. ratio may be varied at will.. 
It should be observed that there must be some slipping since the a.v. 
ratio varies from edge to edge of the belt, the resulting ratio approach- 
ing that of the mean line of the belt. A leather-faced roller might be 
substituted for the belt and a similar series of speeds obtained, the 
cones then turning in the same instead of in opposite directions. 
,Fig. 34 shows, in principle, another form made by the Power and 




Fig. 33. 



30 



ROLLING CYLINDERS AND CONES, ETC. 




Fig. 34. 



Speed Controller Co. Here two equal rollers, C and D, faced with a yield- 
ing material, are arranged to run 
between two equal hollow discs 
A and B. The rollers with their 
supporting yokes (only one of which 
is shown in the elevation) are ar- 
ranged as indicated in the figure 
and are made by a geared connec- 
tion, not shown, to turn opposite 
each other on the vertical yoke 
axes, s. The contour of the hollow 
in the discs must thus be an arc of 
a circle of radius equal that of 
the roller drawn from s as a centre. 
If now the disc B is made fast to 
the shaft, and A, running loose, 
is urged against B by a spring or 
otherwise, a uniform motion of A 
may be made to give varying 
speeds to B by turning the rollers 
as shown. To increase the power 
two sets of discs are often used. 
Fig. 35 shows the Sellers feed-discs used to give a varying a. v. ratio 

between two parallel shafts, one of 

them controlling the feed on a 

machine. 

The two outer wheels are thick- 
ened on their peripheries and run 

between two convex discs BB which 

are constantly urged together by 

hidden coil springs bearing against 

the spherical washers clearly shown. 

The discs BB are supported by the 

pivoted forked arm D. If now the 

disc A be given a uniform a. v., the 

disc C may be made to have a 

greater or less a. v. as the axis of the 

discs BB is made to approach or 

recede from A . 

In Fig. 36 a modified form of 

the Sellers discs used by Jones and 

Lamson is shown. The shaft A is driven by the pulley P and is carried 

by a forked arm supplied with two bearings CC and swinging about a 

point near the centre of the pulley driving P by means of a belt. 




Fig. 35. 



ROLLING OF NON -CYLINDRICAL SURFACED 



31 




Fig. 36. 



The externally rubbing discs B are free to slide axially on the 
shaft A, but turn with it and are 
constantly urged apart by springs 
clearly shown. The internally 
rubbing convex discs are made 
fast to the driven shaft by set- 
screws. To vary the speed of 
D, that of A being constant, it 
is only necessary to vary the 
distance between the shafts. In 
the position shown D has its 
highest speed, the discs rubbing 
at a. When the shaft A is urged 
in the direction of the arrows the 
rubbing radius on B is dimin- 
ished and that on E increased, the discs BB approaching each other. 
The discs BB may be made solid and one of the discs E be urged 
toward the other by a spring on its hub, which would simplify the con- 
struction. 

Grooved Friction - gearing. — Another form of friction-gearing is 
shown in Fig. 37. Here increased friction is ob- 
tained between the rolling bodies by supplying their 
surfaces of contact with a series of interlocking 
angular grooves ; the sharper the angle of the grooves, 
the greater the friction for a given pressure perpen- 
dicular to the axes; both wheels are usually made 
of cast iron. Here the action is no longer that of 
rolling bodies; but considerable sliding takes place, 
which varies with the shape and depth of the groove. 
This form of gearing is very generally used in hoist- 
ing machinery for mines, and also for driving rotary 
pumps; in both cases a slight slipping would be an 
advantage, as shocks are quite frequent in starting 
suddenly, and their effect is less disastrous when 
slipping can occur. 
The velocity ratio is not absolute, but is substantially the same as 
that of two cylinders in rolling contact on a line drawn midway be- 
tween the tops of the projections on each wheel, they being supposed 
to be in working contact. 




Fig. 37. 



50. Rolling of Non-cylindrical Surfaces. — If the angular velocity 
ratio of two rolling bodies is not a constant, the pitch lines will not be 
circular. Whatever forms of curves the pitch lines take, the conditions 
of pure rolling contact should be fulfilled, namely, the point of 



32 



ROLLING CYLINDERS AND CONES, ETC. 




Fig. 38. 



contact must be on the line of centres, and the rolling arcs 

must be of equal length. For example, 
in the rolling bodies represented by Fig. 38, 
with o t and o 2 the axes of rotation, we must 
find the sum of the radiants in contact, 
o ± c+ o 2 c, equal to the sum of any other pair, 
as Oid-\-o 2 e, oJ+o 2 g; and also the lengths 
of the rolling arcs must be equal, cd = ce, 
df = eg. This will cause the successive points 
on the curves to meet on the line of centres, 
and the rolling arcs, being of equal length, 
will roll without slipping. 

There are four simple cases of curves 

which may be arranged to fulfil these conditions : 

A pair of logarithmic spirals of the same obliquity. 
A pair of equal ellipses. 
A pair of equal hyperbolas. 
A pair of equal parabolas. 

We shall also find that any of the above curves may be transformed 
in one way or another and still fulfil the conditions of perfect rolling 
contact, while allowing a wide range of variation in the angular velocity 
ratio. 

51. The Rolling of two Logarithmic Spirals of Equal Obliquity. — 

Fig. 39 shows the development of a pair of such spirals, where, if they 
roll on the common tangent de, 
the axes o t and o 2 will move 
along the lines fg and hk re- 
spectively. The arcs a x c, cb 1} 
etc., being equal to a 2 c, cb 2 , etc., 
and also equal to the distances 
ac, cb, etc., on the common 
tangent, it will be clear that if 
the axes o t and o 2 are fixed, 
the spirals may turn, fulfilling 
the conditions of perfect roll- 
ing contact; for the arc cb x = 
arc cb 2 , and also the radiant 
o l b i + radiant o 2 b 2 = 0/6 -\-o 2 b = 
o x c-\-o 2 c, and similarly for successive arcs and radiants. 

52. To construct two spirals, as in Fig. 39, with a given obliquity. — 
The equation for such a logarithmic spiral is 




Fig. 39. 



ae c 



ROLLING OF TWO LOGARITHMIC SPIRALS. 



33 



where a is the value of r when Q is zero; and b = -7, <f> being the con- 
stant angle between the tangent to the curve and the radiant to the 
point of tangency; and where e is the base of the Naperian logarithms. 

In Fig. 40 let oc = a, and ocd = <f). Taking successive values of 0, 
starting from oc, we may calculate the values of r and thus plot the 
curve. If, however, it is desired to pass 
a spiral through two points on radiants 
a given angle apart, it is to be noticed 
from the equation of the curve that if 
the successive values of d are taken with 
a uniform increase, the lengths of the 
corresponding radiants will be in geo- 
metrical progression. To draw a spiral 
through the points b and e, Fig. 40, bi- 
sect the angle boe, and make of a mean proportional between ob and oe; 
f will be a point on the spiral. Then by the same method bisect foe, 
and find oh; also bisect bof and find ok, and so on; a smooth curve 
through the points thus found will be the desired spiral. 

53. Since these curves are not closed, one pair cannot be used for 
continuous motion; but a pair of such curves may be well adapted to 
sectional wheels requiring a varying angular 
velocity. For example, in Fig. 41, given the 
axes o 1 and o 2 , the angle co x e through which A 
is to turn, and the limits of the a.v. ratio. Make 




Fig. 40. 




c d 

— equal to the minimum a.v. ratio and -^y equal 



o 2 d 



Fig. 41. 



to the maximum a.v. ratio. Then o x e must equal 
Ojd. Now construct a spiral through the points 
c and e. The spiral for B is that part of the 
spiral A constructed about o 1 which would be 
included between radiants o ± b and o x a, equal 
respectively to o 2 c 



and 0$ ( = o 2 d) , which may be found by con- 
tinuing the spiral about o x beyond c or e if 
necessary. Since these curves (Fig. 41) are 
parts of the same spiral, and since by con- 
struction o x c +o 2 c = o x e +o 2 g, A could drive B, 
the points e and g ultimately rolling together 
at d on the line of centres. The conditions 
of rolling contact are evidently fulfilled, as 
will be seen by referring to § 51 . 

54. In Fig. 42 we have a logarithmic 
spiral sector, A, driving a slide, B. Here the driven surface of the 




Fig. 42. 



34 



ROLLING CYLINDERS AND CONES, ETC. 



slide coincides with the tangent to the spiral, the line of centres 
being from o through c to infinity and perpendicular to the direc- 
tion of motion of the slide. In this combination the l.v. of the 
slide will equal the a. v. of A multiplied by the length of the radiant 
in contact, oc. 

55. Wheels using Logarithmic Spirals arranged to allow Com- 
plete Rotations. — By combining two sectors from the same or 
from different spirals, unilobed wheels may be found which may 
be paired in such a way as to fulfil the laws of perfect rolling 

contact. Taking two equal 
sectors from the same spiral, 
we should have a symmetri- 
cal unilobed wheel, as A 
(Fig. 43), and this will run 
perfectly with a wheel B 
exactly like A, as shown. If 
A is the driver, the minimum 
a.v. of B will occur when the 
points d and e are in contact, 
and we shall have 




a.v. B 



o x d 



Fig. 44. 



a.v. A o 2 e' 

The maximum a.v. of B will 
occur when the points / and g 
are in contact. Such wheels 



are readily formed, if the maximum and minimum a.v. ratios are known, 
by the method in § 53, only it is to be noticed that the minimum ratio 
must be the reciprocal of the maximum ratio, and that the angle which 
each sector subtends must be 180°. Unilobed wheels need not be formed 
from equal sectors, in which case the sectors used will not have the 
same obliquity nor will the subtended angles be equal, but the wheels 
must be so paired that sectors of the same obliquity shall be in contact. 
Fig. 44 shows a pair of such wheels in which maximum and minimum 
a.v. ratios occur at unequal intervals; it will, however, be noticed that 
the minimum a.v. ratio must here also be the reciprocal of the maximum 
ratio. 

56. By a similar method wheels may be formed which shall give 
more than one position of maximum and of minimum a.v. ratio; that 
is, we may have either symmetrical or unsymmetrical bilobed wheels, 
trilobed wheels, etc. Fig. 45 shows a pair of symmetrical bilobed 
wheels. Here all the sectors are from the same spiral, all the same 
length, each subtending an angle of 90°. It will be seen that the 



WHEELS USING LOGARITHMIC SPIRALS. 



35 



conditions of rolling contact are perfectly fulfilled, and that if A 

turns uniformly B will have two positions of maximum and two of 

minimum speed. Similarly 

a pair of symmetrical tri- 

lobed wheels could be formed 

where each of the sectors 

subtends an angle of 60°. 

Following the method used 
in obtaining the unsymmet- 
rical unilobed wheels of Fig. 
44, a pair of unsymmetrical 
bilobed wheels could be ar- 
ranged, providing only that 
sectors of the same obliquity 
come into contact and that 
such sectors subtend equal 
angles. Fig. 46 shows a 
pair of trilobed wheels of this form. 

57. Such wheels as those just described cannot be interchangeable , 
but since any two spiral arcs having the same obliquity will roll cor- 
rectly, a unilobe may be made to roll correctly with a bilobe where the 
sectors of the unilobe are from a given spiral and each subtending 180°, 




Fig. 45. 



Fig. 46. 




Fig. 47. 



and where each of the sectors of the bilobe is of the same length 
as one of those of the unilobe, and from a spiral of the same 
obliquity, but where each subtends an angle of 90°. In a similar 
manner a trilobed wheel may be found which could be driven by the 
same unilobed wheel as above, hence also by the bilobed wheel 
found from that unilobe. These wheels would therefore be inter- 
changeable. Fig. 47 shows a set of such wheels which would be 




36 ROLLING CYLINDERS AND CONES, ETC. 

symmetrical wheels. A set of unsymmetrical wheels could be found 
in a similar manner. 

* 58. The Rolling of Equal Ellipses. — If two equal ellipses, each turn- 
ing about one of its foci, are placed in contact in such a way that the 
distance between the axes o x o 2 , Fig. 48, is equal to the major axis of 

the ellipses, we shall find that they 
will be in contact on the line of 
centres and that the rolling arcs 
are of equal length. If the point 
c is on the line of centres o x o 2 , 
we . should have o x e +co 2 = 0^ = 
o x c-\-cd, and therefore cd = co 2 . 
Since the tangent to an ellipse at 
any point, as c, makes equal angles 
with the radii from the two foci, 
o x em = dcn and ecm = o 2 cn; but 
FlG - 48 - since cd = co 2 , the point c is simi- 

larly situated in the two ellipses, and therefore the angle o x cm would 
equal the angle o 2 cn, which would give a common tangent to the two 
curves at c. Hence if o t o 2 is equal to the major axis, the ellipses could 
be in rolling contact on the line o^. Since the distances cd and co 2 , from 
the foci d and o 2 respectively, are equal, it also follows that the arc cf is 
equal to the arc eg which completes the requirements for perfect rolling 
contact. It will also be noticed that the line dee will be straight and 
that a link could connect d and e, as will be seen when discussing link- 
work. 

If A (Fig. 48) is the driver, the a.v. ratio will vary from a minimum 

o h 
when h and k are in contact, and then equal to —-, to a maximum when 

o 2 k 

/ and g are in contact, when it will equal — . The a.v. ratio will be 

unity when the major axes are parallel, the point of contact being then 
midway between o x and o 2 . 

Such rolling ellipses supplied with teeth, thus forming elliptic gears, 
are sometimes used to secure a quick-return motion in a slotting- 
machine. 

59. Multilobed Wheels Formed from Rolling Ellipses. — Non-inter- 
changeable wheels may be formed from a pair of ellipses by contracting 
the angles the same amount in each ellipse. Thus, if the angles were 
contracted to one-half their size, a pair of bilobed wheels could be formed ; 
and if to one- third their size, a pair of trilobed wheels. Such wheels 
would give perfect rolling contact, but could only be used in pairs as 
stated. 



THE ROLLING OF EQUAL PARABOLAS. 



37 



By a different method of contraction a pair of wheels may be formed, 
one of which may be, for example, a 
bilobe and the other a trilobe. By 
this method only parts of the original 
ellipses are used; parts which would 
roll correctly, but which subtend un- 
equal angles in some desired ratio. 
If the arcs subtend angles in proportion 
as 2 is to 3, the angles may be contracted 
or expanded to be 60° and 90°, which 
are in the same ratio, when we shall 
have arcs suitable for a trilobe and a 
bilobe respectively, which will roll cor- 
rectly. For example, assume the foci 
Oj and d (Fig. 49); lay off angles fo x d 
and fde as 2 to 3. Then the point 
/ will lie on an ellipse from which a 
bilobe and a trilobe may be formed by 
contracting the angle fo x d to 60° and 
the angle go 2 d = fde to 90°, as shown 
in the figure. 




Fig. 49. 



6o. The Rolling of Equal Parabolas.— Two parabolas may be con- 
sidered as two ellipses with one focus of each removed to infinity. In 




at infinity 



e at infinity 



Fig. 50. 



the ellipses of Fig. 48 suppose the foci o 1 and e to be so removed; we shall 
have the parabolas of Fig. 50 in contact at the point c and in perfect 



38 



ROLLING CYLINDERS AND CONES, ETC. 



rolling contact, one turning about its focus o 2 as an axis, and the other 
having a motion of translation perpendicular to o t d. 

To prove the rolling action perfect, assume the parabolas with their 
vertexes in contact at ra. Let / be the point on the turning parabola 
which will move to c, so that o 2 f=o 2 c. Draw fg parallel to o 2 c, and since 
the parabolas are equal we shall have lg=o 2 f, therefore lg=o 2 c; but 
since o 2 k is the directrix of the parabola whose focus is now at I, lg=gk; 
therefore gk = o 2 c, and as this parabola slides perpendicular to o x d, the 
point g would also move to c. The rolling arcs mf and mg are equal. 
Thus the parabola turning about o 2 would cause the other parabola to 
have translation perpendicular to o^d, the two moving in perfect rolling 
contact. 

61. The Rolling of Equal Hyperbolas. — If two equal hyperbolas are 
placed as in Fig. 51, so that the distances between their foci o x and o 2 , 
and d and e, are each equal to fg = hk, the distance between the vertexes 
of the hyperbolas, we shall find them in contact at some point c. If 




Fig. 51. 

the foci o 1 and o 2 are then taken as axes of rotation, the hyperbolas will 
turn in perfect rolling contact. To prove this take the point I on the 
hyperbola whose foci are at o x and d so that o 1 l=dc and o 1 c=dl. Then 
since a tangent at any point on a hyperbola makes equal angles with 
the radii from the two foci, the tangent at I will bisect the angle o t ld 
and the tangent at c will bisect the equal angle o t cd. If now the branch 
0J1I is placed tangent to the branch dkc with the points I and c in con- 
tact, the radius lo t must fall on o x c and dl on dc. Since the difference 



THE ROLLING OF EQUAL HYPERBOLAS. 39 

between the radii from the two foci to any point on a hyperbola is a 
constant and equal to the distance between the vertexes, o^ — dc^hk; 
but o t l was taken equal to dc, hence o x c — o x l^hk. Then, since o x o 2 was 
originally assumed equal to hk, we shall have 0^ — 0^ = 0^, and there- 
fore the line o x o 2 c will be a straight line, and the point of contact c will 
lie on the line of centres. The arc Ih which is equal to ck will also be 
equal to cf. Therefore the hyperbolas will be in perfect rolling contact. 
The same reasoning will apply for any position of the point of contact. 
It is interesting to note that since o 1 o 2 =de = a constant, and o x d=o 2 e = a 
constant, the linkage o 1 o 2 ed with the axes o ± and o 2 fixed would cause the 
same a.v. ratio about o x and o 2 as the rolling hyperbolas would give. 

If the hyperbola turning about the axis o 2 is the driver, the a.v. ratio 
will be a minimum when the vertexes / and k are in contact and will 

be —~ ; this ratio will increase until the point of contact is at infinity, 

x n> 

when the ratio would be unity, and would correspond to the position 
of the linkage when o 1 o 2 and de are parallel. Further rotation would 
bring the opposite branches of the hyperbolas into contact, the maxi- 
mum a.v. ratio occurring as the points g and h come together, when its 

value becomes ~. The construction shown in the figure will allow 
0J1 G 

only a limited motion. 



CHAPTER V. 

CONNECTION BY BANDS OR WRAPPING CONNECTORS— BELTS, CORDS 

AND CHAINS. 

62. A flexible wrapping connector may be paired by force-closure 
with a pulley, and two such pairs may be combined to connect two 
axes, whether parallel, intersecting, or neither parallel nor intersecting. 
Flexible connectors may be divided into three general classes: 

1° Belts made of leather, rubber, or woven fabrics are flat and thin, 
and require pulleys nearly cylindrical with smooth surfaces. Flat ropes 
may be classed as belts. 

2° Cords made of catgut, leather, hemp, cotton, or wire are nearly 
circular in section, and require either grooved pulleys or drums with 
flanges. Rope gearing, either by cotton or wire ropes, may be placed 
under this head. 

3° Chains are composed of links or bars, usually metallic, jointed 
together, and require wheels or drums either grooved, notched, or toothed, 
so as to fit the links of the chain. For convenience the word band may 
be used as a general term to denote all kinds of flexible connectors. 

Bands for communicating continuous motion are endless. 

Bands for communicating reciprocating motion are usually made fast 
at their ends to the pulleys or drums which they connect. 

63. The Line of Connection of a pair of pulleys connected by a band 
is the neutral line or axis of that part of the band which transmits the 
motion. The neutral line of the band is a line which is neither com- 
pressed nor stretched in passing around the pulley. 

The Pitch Surface of a pulley over which a band passes is the surface 
to which the line of connection is always tangent; that is, an imaginary 
surface whose distance from all parts of the acting surface of the pulley 
is equal to the distance from the acting surface of the band to its neutral 
line. Belts are commonly used to transmit a nearly constant and con- 
tinuous velocity ratio, and in this case the acting surfaces are cylindrical. 

The Effective Radius of a pulley is the radius of its pitch surface. 

The Pitch Line of a pulley is the line on its pitch surface in which 

40 



VELOCITY RATIO. 



41 



the centre line of that part of the band which touches the pulley lies. 
The line of connection is tangent to the 'pitch line. 

64. Velocity Ratio. — If we assume the band to be inextensible, and 
that there is no slipping between it and the pulley, the l.v. of the pitch 
surfaces of the connected pulleys must be equal. Let D x and D 2 be 
the diameters of the connected pulleys ,and N t and N 2 their revolutions 
per minute respectively, the distance from the surface of the pulley 
to the neutral axis of the belt being p ; then 

7tN 1 (D l +2p) =7iN 2 (D 2 +2p).. 
. D 1 +2p _N 2 

N< 



(15) 



'.' D 2 +2p 

That is, the angular velocity ratio, as in rolling cylinders, is inversely 
proportional to the effective diameters, and is constant for circular 
pulleys. As the thickness of thin flat bands or belts is generally small 
compared with the diameters of the pulleys connected, it may be neg- 
lected, and the acting surfaces of the pulleys may then be considered 
to have the same linear velocity. The approximate angular velocity 
ratio will then be given by the equation 

|^ ; . ......... a«, 

Belts and cords are not suited to transmit a precise velocity ratio r 
because they are liable to stretch or to slip on the pulleys. This freedom 
to stretch and slip is an advantage in powerful and quick-running 
machinery, as it prevents shocks which are liable to occur when a machine 
is thrown suddenly into gear, or when there is a sudden fluctuation in 
the power transmitted. 

Belts or Thin Flat Bands. 
65. Crowning of Pulleys. — If we suppose a belt to run upon a revolv- 
ing conical pulley, it will tend to lie flat upon the conical surface, and, 
on account of its lateral stiffness, will 
assume the position shown in Fig. 52. 
If the belt travels, in the direction of the 
arrow, the point a will, on account of 
the pull on the belt, tend to adhere to 
the cone and will be carried to b, a point 
nearer the base of the cone than that 
previously occupied by the edge of the 
belt: the belt would then occupy the 
position shown by the dotted lines. 
Now if a pulley is made up of two equal 
cones placed base to base, the belt will tend to climb both, and would 
thus run with its centre line on the ridge formed by the union of the 



A 




Fig. 52. 



42 



CONNECTION BY BANDS OR WRAPPING CONNECTORS. 



r= 


G 


=3/ 


i ° 


°" 


cC 


£ 





A B 

Fig. 53. 



two cones. In practice pulley rims are made crowning, except in cases 
where the belt must occupy different parts of the same pulley. In 
Fig. 52 two common forms of rim sections are shown at C and D; that 
shown at C is most commonly met with, as it is the easier to construct. 
When pulleys are improperly located, the belt will generally work 
toward the position where it is tightest, or will run toward the high side 
of the pulley; this is due to the lateral stiffness of the belt, and could 
be explained in the same way as the climbing on a conical pulley. 

66. Tight and Loose Pulleys are used for throwing machinery into 
and out of gear. They consist of two pulleys placed side by side upon 

the driven shaft CD (Fig. 53); A, the tight pulley, is 
keyed to the shaft; while B, the loose pulley, turns 
loose upon the shaft, and is kept in place by the hub of 
the tight pulley and a collar. The driving-shaft carries 
a pulley G, whose width is the same as that of A and B 
put together, or twice that of A. The belt, when in 
motion, can be moved by means of a shipper that guides 
its advancing side, either on to the tight or the loose 
pulley. The tight pulley is sometimes made larger than 
the loose pulley, so that the belt may be slack when it 
is on the loose pulley. In such a case the loose pulley 
has a flange upon the edge next the tight pulley, of a 
diameter equal to that of the tight pulley; this flange aids in the transfer 
of the belt from one pulley to the other. 

The acting surface of a pulley is called its face; a pulley with a six- 
inch face is one having a face six inches wide. The faces of pulleys are 
always made a little wider than the belts which they carry. The pulley 
G (Fig. 53) has a flat face, because the belt must occupy different posi- 
tions upon it; while A and B have crowning faces, which will allow the 
shifting of the belt, and will retain it in position when shifted upon 
them. 

67. Length of Belts connecting Parallel Axes. — Pulleys for belts 
connecting parallel axes usually have their pitch circles in one plane, 
which is perpendicular to the axes. «^ 

The belts may be crossed as in 
Fig. 54, where the pulleys rotate 
in opposite directions, or open as 
in Fig. 55, where the pulleys rotate 
in the same direction. 

Crossed Belts. — Let D and d 
(Fig. 54) be the diameters of the 
connected pulleys; C the distance between their axes 
the belt. Also let 

(D+d) = J and (D-d) = J. 




Fig. 54. 



L the length of 



LENGTH OF BELTS CONNECTING PARALLEL AXES. 



43 



Then 



L = 2{mn-\- no-\- op) 
= (I + 0W2C cos 6+ (f + #)<* 

= ^ + 0^+2Ccos0, . . . 



(17) 



where 



at an+bo D+d I 



sintf = — j-= 
ao 



Open Belts. — Using the same 
notation as for crossed belts, we 
have (Fig. 55) 



L = 2(mn+ no-\- op) 



= (~ + 0)D+2Ccosd+ 




Fig. 55. 



(H< 



=^(D+d) + d(D-d) + 2C cos 



-here 



and 



=^I+6A+2C cosd, 



sin d 



cos 



an — 

C 


bo D-d 
2C 


A 
= 2C 


4- 


A 2 

4C 2 ' 





(18) 



For an open belt, d is generally small, so that # = sin 6, very nearly; 
then 

L=* r r+£ 1 +2C y Jl-~, nearly, 



2C 



A 2 I A 2 ) 

\ i^+^j 1-4^2 [> nearly. 



4C' 



If we expand the quantity under the radical sign, and neglect all 
terms having a higher power of C than the square in the denominator, 
as C is always large compared with A, we have 

A 2 , . A 2 

8C 2 



L=%Z+2C 



UC 2+1 



...J, 



or 



7T J 2 

L = -J+2C+^, very nearly. 



(19) 



44 



CONNECTION BY BANDS OR WRAPPING CONNECTORS 



68. Speed-cones are contrivances for varying and adjusting the 
velocity ratio of a pair of rotating parallel axes by means of a shifting 
belt: they may be continuous cones or conoids, as in Figs. 56 and 58, 
where the velocity ratio can be varied gradually while they are in motion 
by shifting the belt, or sets of stepped pulleys, as in Figs. 57 and 59. 

In order that the belt may be equally tight in all positions, the length 
L, previously found, must not vary. 

Crossed Belts (Figs. 56 and 57). — Here (equation 17) L depends only 
on the distance between centres C, the sum of the diameters I , and the 
angle 0, which is itself dependent on C and I '. As C is a constant, the 
axes being parallel, therefore the sum of the diameters, (D+d)=I, in 




Fig. 56. 



Fig. 57. 



Fig. 58. 



Fig. 59. 



all positions of the belt must be constant in order that the length of the 
belt L may not vary. This condition is fulfilled by two cones with equal 
angles at their apexes, as in Fig. 56, or a pair of stepped pulleys having 
the sum of the diameters for each pair of steps a constant. 

Open Belts (Figs. 58 and 59). — Here the length of the belt is as found 
in equation (19), 



fj+*+£- 



(20) 



Hence, if an open belt is to run on a pair of pulleys, the sum and dif- 
ference of whose diameters are I a and J a , and the same belt is also to 
run on another pair of pulleys, the sum and difference of whose diame- 
ters are 2 X and J x , we have, since the length of the belt must be the 
same in both cases, 



£J +2C+ 



AC 2 



2Vf2C+ 



4C 



from which 



**->?*? 2 7zC ' 
D. 



(21) 



From this equation and the equation —^=— ?, where N represents the 

r.p.m. of the driving shaft and n x the r.p.m. of driven shaft when the 
belt is on the diameters D x , d x , the diameters D x and d x could be 



SPEED-CONES. 45 

found. It is, however, generally accurate enough to solve equation 
(21) for I w by substituting for A x , which should be D x —d x for the 
open belt, its value found by assuming the belt crossed; that is, solv- 
ing for D x —d x , assuming I x = I a \ thus 

D x = n x 

d x N' 
D x —d x _n x —N 

n —N 
and A x (for a crossed belt) = x AJ I a (22) 

ftxi -iV 

Substituting this in equation (21) we have 

■N 



2 j ^-^V y V 

Ja \n x +N 2a \ 



2*- 1 '* ZzC (23) 

D ft 
From this value of l x and the fundamental relation -~r =Tf we may 

d x JN 

calculate the values of the diameters D x and d x . 

Hence the process of designing a set of speed pulleys is thus : Having 
given JV, the r.p.m. of the driving shaft, and C, the distance between 
axes, decide on the speeds n 1} n 2 , n 3 , etc., of the driven shaft; choose either 
diameter of any pair of steps, and calculate the other diameter by equa- 
tion (16), thus finding values for l a and A a . From these the diameters 
of any other pair of pulleys can be found. In the case of a crossed belt, 
the other diameters are found by using the two equations 

D a -\- d a = D x -\- d x = a constant 
and 

D x _ftx 

d~N' 

In the case of an open belt find values for A 1} A 2 , A 3 , etc., assuming 
the belt crossed as in equation (22). Substituting these values in equa- 
tion (21) will give sufficiently accurate values for 2\, I 2 , I 3 , etc. From 
these values of I the successive pairs of diameters may be found. 

In the case of conoids (Fig. 58), for an open belt, a series of equi- 
distant diameters could be calculated for each, and curves drawn through 
the ends of these diameters would give the shapes for the conoids. 

In designing a pair of speed-cones or pulleys, it is very often desira- 
ble to have them both alike so that they can be made from one pattern, 
the velocity ratios being thus fixed. The diameters of both pulleys, as 
in Figs. 56 and 58, would be the same for the central position of the 
belt; in Figs. 57 and 59, when the number of steps is odd, the middle 
diameters will also be the same, or D 2 = d 2 ; then A=D 2 —d 2 =0, and 
Z=D 2 +d 2 =2D 2 . 



46 CONNECTION BY BANDS OR WRAPPING CONNECTORS. 

69. Suppose we wish to construct two equal speed-cones (Fig. 60) 
to give to the driven shaft a range of speed 
between r^ and n n r.p.m., n n being greater 
— ' than n v Let the smallest diameters D^ = d n 
be given; it is necessary first to determine the 
largest diameters D n = d 1 , and N, the r.p.m. 
of the driving shaft. Assuming the belt on 

Pi 
d t 




D v d lf we may write ~=^; and assuming 



Fig. 60. 



the belt on D n , d n would give ,- = 

d n 



N' 



but 



since D 1 = d n and D n = d x we have 



N' 



N 



or 



N=V 



n.rir 



(24) 



Knowing the value of N the large diameters D n = d 1 are readily found. 
If we assume the smallest diameters D 1 = d n = 4", n n = 600 r.p.m., 
and 7^ = 60 r.p.m., i\T will be found to be 189.7 r.p.m. and the largest 
diameters, D n = d 1 , will be 12". 65. 

If the belt were crossed, the cones shown in Fig. 60 would answer. 

In this case the middle diameters would be equal to — ^— — 1 = 8 /, .S2; or 

if a stepped pulley having n steps were desired, divide op into (n — 1) 
parts, and erect perpendiculars D 2 , D 3 , etc., which would give the diam- 
eters of the successive steps; then draw in the pulley as shown by the 
dotted lines. 

If the belt is open, the largest and smallest diameters are found as 
above. To find the middle diameters D m , d m , which are equal since the 
pulleys are to be alike, equation (21) is used, I a being Dj-Mi or D n +d n ; 
A a being D l —d l or D n — d n ; and d x being the difference between the 
middle diameters D m —d m , which difference is zero since D m =^d m . Mak- 
ing these substitutions in equation (21) we have 



2™ = ^!+ 



2izC 



Using the same data as taken above with the crossed belt, and letting 
the distance between the axes C = 20", we find 



74 82 
16.65+^^ = 17 ,, .25, 



125.66 



and 



D 7 



8 ,r .65. 



Having thus determined the middle diameter, the curve of the conoid 
may be an arc of a circle passing through the extremities of the three 



EFFECTIVE PULL. 47 

diameters, as in Fig. 58. This middle diameter would also be correct 
for the middle step of any stepped pulley having an odd number of steps. 
Other steps might be found graphically by use of the principle of Fig.. 60, 
the curve being used in place of the straight line; the results can be 
checked by calculation. 

As a flat belt tends to climb a conical pulley, it is necessary, when 
speed-cones or conoids are used as in Jigs. 5Q and 58, to provide a 
shipper which will guide the advancing side of the belt for each pulley 
and retain it in its proper place; the guiding forks are moved simul- 
taneously, and are placed as near each pulley as possible. 

If the distance between the axes of the pulleys to be connected by 
an open belt is considerable, as is often the case in belting from sn, 
overhead shaft to a machine, or if, is as sometimes the case, one of the 
axes is adjustable (the proper tension on the belt being obtained by 
the weight of the pulley on the adjustable axis), the diameters can be 
calculated as though the belt were crossed. However, the distance 
between the axes may be large, and yet the range of speed may be so 
great, that with a fixed distance between the axes, an open belt would 
not be sufficiently tight toward the middle diameters if it were properly 
tight at the ends. In this case the diameters would need to be found 
as in the solution already given for the open belt. 

70. Effective Pull. — By the effective pull on a belt we mean the pull 
that is doing the work, that is, the difference in the tension of the two 
sides of the belt. If in Fig. 55 we assume A the driver .and B the fol- 
lower, 2\ the tension in the tight side and T 2 the tension in the loose 
side of the belt, the effective pull will be (T 1 — T 2 )=P. Suppose A 
to be 30" in diameter, to make 200 revolutions per minute, and to 
carry a belt transmitting 4 horse-power or 4X33,000 ft.-lbs. per minute; 
what is the effective pull on the belt? 

Here the work done by the belt, that is, the effective pull on the belt, 
in lbs., multiplied by its speed in feet per minute, must equal the horse- 
power transmitted multiplied by 33,000, or 

30 

^2X200X3,1416XP = 4X33,000. 

.*. P = 84.03 + lbs. 

From the above example it can readily be seen that the quicker a belt 
travels the smaller the pull for a given horse-power transmitted. 

Belts and Pulleys to Connect Non- parallel Axes. 

71. The plane of a pulley is a central plane through the pulley per- 
pendicular to its axis, or is the plane of the pitch line of the pulley. The 
point of delivery of a pulley is the point in the pitch line at which the 
belt leaves the pulley. 



48 CONNECTION BY BANDS OR WRAPPING CONNECTORS 



The general principle governing the arrangement of belt-pulleys upon 
non-parallel axes may be stated as follows: The belt must always be 
delivered in the plane of the pulley toward which it is running. That 
is, a belt leaving a pulley may be drawn out of the plane of the pulley; 
but when approaching a pulley, its centre line must lie in the plane 
of the pulley. 

Guide-pulleys are used to change the directions of belts, and are 
placed according to the above rule. It is possible, by the use of two 
guide-pulleys, to connect any pair of pulleys by an endless belt, and 
the guide-pulleys may be so placed that the belt will travel in either 
direction, which is sometimes a great advantage. 

72. General Case. — Let A A and BB (Fig. 61) be the vertical and 
horizontal projections of a pair of pulleys to be connected by a flat belt, 

their planes intersecting on SS. 
Choose a and c convenient points 
on the trace SS; from a draw a 
line ab, and from c draw cd, both 
in the plane of A and tangent 
to the pitch line of A; similarly 
draw ae and cf in the plane of 
B. The two lines ab and ae deter- 
mine the plane of the guide-pul- 
ley C, it being so placed that its 
pitch line is tangent to both ab and 
ae; in this position it can take a 
belt delivered either at e or b, as 
both e and b are in its plane; it can 
also deliver a belt to A or B, as it 
has its points of delivery in the planes of A and B. In the same man- 
ner, the two lines cd and cf in the planes A and B respectively fix the 
position of D. Having placed C and D, the belt may be put on as 
shown; the same side of the belt comes in contact with all of the pul- 
leys, and it is immaterial which way it runs. 

As double bending, i.e., bending back and forth, tends to injure a 
belt, it is desirable, when possible, to have the same side of the belt 
come in contact with each of the pulleys. 

73. Quarter- turn Belt.— When a belt always travels in the same 
direction, it is only necessary to provide for its running properly in that 
direction, and in such a case the belt must always be delivered in the 
plane of the pulley toward which it is running. 

Fig. 62 shows a quarter-turn belt connecting two pulleys A and B 
whose axes are in parallel planes and at right angles with each other. 
SS is the trace of the intersecting planes of A and B, and if we suppose 




Fig. 61. 



QUARTER-TURN BELT. 



49 



the pulley A to revolve in the direction indicated by the arrow, it de- 
livers the belt at a in the plane of B, while B delivers 
the belt in the plane of the pulley A; thus the ro- 
tation indicated by the arrows is allowable. If, how- 
ever, we attempt to turn A in the opposite direction, 
the belt will immediately leave the pulleys, for d, the 
point of delivery of B, is not in the plane of A, nor is 
€ in the plane of B. If the pulley B with its plane is 
swung about SS as an axis into any position such as 
B v the belt will still run in the direction indicated by 
the arrow, as the same conditions exist as before. In 
fact, if we draw on a piece of paper the pitch lines of 
two pulleys as A and C, and draw a line SS tangent to 
them, the paper may then be folded on SS, and it 
can easily be seen that the point of delivery a of 
the pulley A is in the plane of C, and that the point 
•of delivery of C is in the plane of A , no matter how 
the plane of C is turned about SS. 

It is not well to use quarter-turn belts where 
there is any liability of the motion being reversed, 
as when this happens they will immediately leave the pulleys. The 
angle at which the belt is drawn off from the pulleys should not be 
great, since when this is the case the belt is much strained, and does 
not hug the pulley well; power is also lost in side-slipping. To make 
this angle as small as possible, the pulleys should be placed a sufficient 
distance apart, and they should be as small in diameter and carry as 
narrow a belt as practicable. 

Fig. 63 shows a quarter-turn belt arranged with a double pulley 
so as to run in either direction; A is the driving pulley and B is the 




Fig. 62. 



i 



T 3, 




m 



feES 



Fig. 63. 



driven pulley, which is designed for two positions of the belt. For 
the rotation in the direction of the full arrow the belt is drawn in full 
lines; for the opposite rotation the belt is drawn dotted. This arrange- 
ment was made use of on a small moulding-machine where the spindle 
attached to B could be made to turn in opposite directions, its cutters 
being made to worb when rotating in either direction. 



50 CONNECTION BY BANDS OR WRAPPING CONNECTORS. 




Two shafts at right angles to each other are often connected by 

means of a belt and one guide- 
pulley, as shown in Fig. 64. Here 
ab is the trace of the two inter- 
secting pulley planes, and the 
plane of the guide-pulley C is 
found by taking a point in ab, as 
c, and from it drawing lines ce 
and cd tangent to the pitch lines 
of A and B respectively, which 
fix the plane of C. This arrange- 
ment is often used to drive mill- 
stones, which are carried by the 
upright shafts of the pulleys like A ; the faces of the upright shaft pulleys 
are made straight and considerably wider than the belt, so as to allow 
a slight motion up or down to adjust the stones, or a movement of the 
guide-pulley C to tighten the belt, which then merely runs on a different 
part of the pulley A. 

The shaft A might be arranged to carry a tight-and-loose pulley, and 
the belt might be shifted by moving the guide-pulley C, the motion of 
which should be such that the belt has the proper tension in each posi- 
tion. Whenever a belt is used in the above manner, the working part 
of the belt should pass from a to b, and not over the guide-pulley C ; this 
should be true for all cases where one guide-pulley is used, as the extra 
friction, due to the working pull, is saved on the guide-pulley bearings. 

Figs. 65 and 66 show another method of connecting two shafts at 
right angles, two guide-pulleys being used in each case. The belt can 





Fig. 65. Fig. 66. 

travel in either direction, and the same side of the belt comes in contact 
with each pulley, except in the case of D (Fig. 66), where it is not pos- 
sible on account of the crossed belt. It will be noticed that the directional 
relation between A and B is the same in Figs. 65 and 66. 

When a crossed belt is used to connect two pulleys, as in Fig. 54, 



QUARTER-TURN BELT. 



51 





Fig. 67. 



Fig. 68. 



it is necessary to give the two straight parts of the belt a half-twist to 

have them pass each other, and to bring the same side of the belt in 

contact with each pulley. 

74. Let A and B (Fig. 67) be two pulleys whose axes are parallel, 

but whose planes do not coincide. Place the guide-pulleys C and D r 

whose diameters are equal 

to the distance between the d hd h (C 

planes of the pulleys, and 

whose planes are parallel to 

the axes of, and tangent to, 

the pulleys, as shown in the 

figure. The belt can then 

be applied as indicated, and 

can run in either direction. 

This arrangement could also 

be used to connect two axes 

quite near each other, and 

thus obtain a long belt, which works much better than a short one. 
It is generally more convenient to place the guide-pulleys C and D 

on the same axis : such an arrangement is shown in Fig. 68, where the 

guide-pulley C is in a plane perpendicular to the common axis of the 

guide-pulleys and tangent to B, while D is in a plane parallel to that 

of C and tangent to the pulley A. In this case, the belt can run only in 

the direction indicated by the arrows. 

We may now imagine the pulley B to be swung around, its plane 

still remaining tangent to C and D, and its axis in a plane parallel to 

that of the axis of A : the method of arranging the guide-pulleys remains 

the same. This arrangement is often used to connect axes in the same 

horizontal plane, or in two horizontal planes one a little above the 

other; the guide-pulleys are then placed on a common vertical shaft on 

which they turn loosely, being held in position by collars properly placed. 

The name mule-pulleys is often applied to guide-pulleys arranged, as 

indicated above, upon a vertical shaft. 

Fig. 69 shows, in elevation, two shafts at right angles to each other 
connected by means of a belt and mule-pulleys. 
In order that a belt may run properly on pul- 
^ leys having other than horizontal axes, they 
are made more crowning, and, when on per- 
pendicular axes, have flanges on their lower 
edges; or, better, stationary flanges similarly 
placed, and carried by the shaft on which the 
pulleys revolve. The proper radial section 

for the guiding flange is shown at A, where a projecting lip serves 

to guide the belt when approaching the mule-pulleys, and the recess 




=^ 



Fig. 69. 



52 CONNECTION BY BANDS OR WRAPPING CONNECTORS 



behind the lip permits the belt to lie flat on the pulley. A straight 
flange is liable to cause the belt to climb and strain its edge. 

Fig. 70 shows a method of connecting two hori- 
zontal axes making an angle with each other. The 
perpendicular distance between the axes must be a 
little greater than the sum of the radii of the pulleys 
A and B. In this case, A and B are of the same size, 
and both guide-pulleys can be placed on the same 
shaft and will revolve together; the belt may also 
travel in either direction. A might drive D, and 
B and C then act as guide-pulleys. 

75. Binder-pulleys. — Guide-pulleys are often used, 
as in Fig. 64, to increase the arc of contact of the belt, 
and also, as there, to tighten the belt; when so used, 
they are called binder- or tightening -pulleys, and are 
always applied to the loose side of the belt. 

Pulleys for belts could be combined in many other 
ways, but the same principles govern the arrange- 
ment in each case. When convenient it is best to 
arrange the belt to travel in either direction, as engines 
are sometimes moved backward a part of a turn, thus 
Fig. 70. rendering any belt not admitting of motion in either 

direction liable to be thrown off. 



Cords and Ropes. 

76. Pulleys for round ropes or cords must be provided with V grooves 
to keep the ropes in place; any flat bands not having sufficient lateral 
stiffness would also require flanges to keep them on the pulley. 

Cords of small diameter are used to transmit small amounts of power, 
as for driving the spindles in spinning-machinery. 

Hemp and cotton ropes are now very generally used to transmit 
quite large amounts of power ; when so used they are often run in sets, 
each pair of pulleys carrying several ropes. The grooves are made 
V-shaped, and the ropes are drawn into them, thereby increasing the 
hold upon the pulleys. Rope pulleys are usually made of cast iron, 
and the grooves are turned. Fig. 71 A shows a section of the rim of 
a pulley for carrying a number of ropes. 

Large amounts of power are now successfully transmitted over 
long distances by rapidly moving wire ropes carried by large, grooved 
wheels. Wire ropes will not support without injury the lateral crushing 
due to the V-shaped grooves; hence it is necessary to construct the 
pulleys with grooves so wide that the rope rests on the rounded bottom 
of the groove, as shown in Fig. 71 B, which shows a section of the rim 
of a wire-rope pulley. The friction is greatly increased, and the wear 



DRUM, OR BARREL. 



53 



of the rope diminished, by lining the bottom of the groove with some 
elastic material, as gutta-percha, wood, or leather jammed in tight. 





Fig. 71. 
Cords and ropes, especially wire ropes, are in general only used to 
connect parallel axes, but when otherwise used the same rules as were 
given for flat bands govern the arrangement. As ropes have no lateral 
stiffness, they are not used to connect vertical axes without supplying 
guide-pulleys to insure the proper guiding of the ropes into the grooves. 

77. Drum, or Barrel. — When a cord does not merely pass over a 
pulley, but is made fast to it at one end, and wound upon it, the pulley 
usually becomes what is called a drum, or barrel. A drum for a round 
rope is cylindrical, and the rope is wound upon it in helical coils. Each 
layer of coils increases the effective radius of the drum by an amount 
equal to the diameter of the rope. A drum for a flat rope has a breadth 
oqual to that of the rope, which is wound upon itself in single coils, 
each of which increases the effective radius by an amount equal to the 
thickness of the rope. 

78. Small cords are often used to connect non-parallel axes, and 
very often the directional relation of these axes must vary. The most 
common example is found in spinning frames and mules, 
spindles are driven by 
cords from a long, cylin- 
drical drum, whose axis 
is at right angles to the 
axes of the spindles. In 
such cases, the common 
perpendicular to the two 
axes must contain the 
planes of the connected 
pulleys ; both pulleys 
may be grooved, or one 
may be cylindrical, as in 
the example given above. 
Fig. 72 shows two 

grooved pulleys, whose axes are at right angles to each other, connected 
by a cord which can run in either direction, provided the groove is deep 



where the 




54 CONNECTION BY BANDS OR WRAPPING CONNECTORS. 



enough. To determine whether a groove has sufficient depth in any 
case, the following construction (Fig. 73) may be used. Let ab and a'b r 
be the projections of the approaching side of the cord; pass a plane 
through ab parallel to the axis of the pulley; it will cut the hyperbola. 
cbd from the cone feg, which forms one side of the groove. The cord 
will lie upon the pulley from b to i, where it will leave the hyperbola 
on a tangent. If the tangent at i falls well within the edge of the 
pulley c, the groove is deep enough. It will usually be sufficient to 
draw a straight line, as ab (Fig. 72), and see that it falls well inside 
of the point corresponding to c in Fig. 73. 

Gearing-chains. 
79. In cases where a considerable amount of power has to be trans- 
mitted between two shafts running at a slow speed, metal chains, called 
gearing-chains or pitch-chains, are used with toothed wheels or sprocket- 
ivheels. When a chain is to drive or be driven by a sprocket-wheel, the 
acting surface of the wheel must be adapted to the figure of the chain, 
so as to insure sufficient hold between them. Guide-sheaves for chains 
are made circular, and grooved to suit the form of the chain. Chain 
drums or barrels have one end of the chain made fast to them, and the 
chain is guided by a suitably formed helical groove. Such drums are 
used in cranes. 

Gearing-chains are usually made of flat links, riveted or pinned 
together at their ends, so as to allow a free turning of the links at the 
joints. The most simple form consists of double and single links arranged 
alternately, the single link being made thick enough to have the same 
strength as the double links. 

A wheel for such a chain (Fig. 74) has a polygonal prism abc for its 
pitch surface, and when the links are of the same length, the pitch line 

is a regular polygon; each side 
of the polygon is equal to the 
effective length of a link, or 
the distance from centre to 
centre of the rivets. The teeth 
of the wheel are arranged to 
correspond to the chain, and 
are alternately single and dou- 
ble: the single teeth pass be- 
tween the two thin flat parts 
of the double links, and the 
double teeth pass on both sides 
of the thicker single links. On 
larger wheels, the double teeth 
are sometimes left off, and their 
places are supplied by short projecting lugs, which serve to keep the chain 




Fig. 74. 



GEARING-CHAINS. 



55 



in position sidewise. The acting parts of the teeth outside of the pitch 
line are made up of arcs of circles, whose centres are the adjacent angles 
of the pitch polygon; thus the arc de is drawn with b as a centre, and 
jg with c as a centre. The space between the teeth is shaped to receive 
the chain, as dmn. The length of the teeth should be such that they 
come up to, or project a little beyond, the outer edge of the chain when 
it is in position on the chain-wheel. 

The sprocket-wheel may be constructed without the double teeth wn 
when the links are short and the single and double links may have dif- 
ferent lengths, the latter being longer to give sufficient strength to the 
tooth or sprocket. In such case the wheel is made a little thinner than 
the single links, its teeth decreasing in thickness toward their points to 
better enter the chain. The pitch line is a polygon having alternate 
long and short sides, and the pitch is usually taken as the combined 
length of a long and short link. This form of a chain (known as a block 
chain) and wheel, of one inch pitch is now very accurately and econom- 
ically made of hardened steel and is extensively used on bicycles. 

A gearing-chain is sometimes made up of two systems of links, sepa- 
rated by an enlargement of the pins connecting 
the successive links, as shown in Fig. 75. The 
sprocket-wheel is placed in the space between the 
two systems of links, and the teeth gear with the 
enlarged portions of the pins connecting the links; 
their acting surfaces are arcs of circles, as in the 
preceding case. 

Light chains of this form are now extensively 
used on automobiles and are supplied with rollers 
to reduce friction and wear. Fig. 7G shows a roller-chain as made by the 

Whitney Mfg. Co. The rollers R turn 
on the hollow sleeves S supplied with 
flats at their ends to keep them from 
turning in the inner links. The rivets 
P are fast in the outer links L and 
turn on the inside of the sleeves S. 

Fig. 77 shows a wheel for a chain having oval 
links; here the pitch line is a polygon, with alter- 
nate long and short sides; the pitch of the link 
lying flatwise is equal to its longer diameter plus 
the diameter of the link iron, while that of the link 
standing edgewise is equal to its longer diameter 
minus the diameter of the link iron. In this case, 
the short sides of the polygon only are provided 
with teeth, which act on the ends of the flat-lying 
links, and receive those standing edgewise between them. 






Fig. 77. 



56 CONNECTION BY BANDS OR WRAPPING CONNECTORS. 



Chain-wheels for oval chains, which are the most common, are 
often made with a groove to receive the links standing edgewise, 
and a series of pockets into which the links lying flatwise fit. This 
form of wheel is often used to exert a pull on a chain passing partially 
around it, as shown in Fig. 78. Here the pull is a downward one 
on the chain X, which passes under the pocketed chain-wheel A, and 
over a guide-sheave E, from which it passes downward, and is deposited 
in a box. A chain-guide, C, is placed under the wheel A, to insure the 




Fig. 78. 

proper pocketing of the chain and guidance to the sheave E; this guide 
is provided with a groove in which the links standing edgewise move, 
and it is placed far enough away from the chain-wheel to allow the links 
to move freely when properly pocketed. A chain-stripper, D, passing 
into the groove of the wheel A, removes the chain from the wheel at 
the proper place: its action can be clearly seen from the figure. A pro- 
longation of the stripper D covers 
the guide-sheave E, and insures the 
proper passing downward of the 
chain. 

The chain could be arranged to 
pass over a chain-wheel: it would 
then exert an upward pull. In such 
cases, the guide-sheave is dispensed 
with, and a guide is placed at the 
point where the chain comes in con- 
tact with the wheel to insure the 
proper pocketing of the links. This 
arrangement is often used in small 
hoisting-machines, the weight being directly lifted by the chain which 
passes over the chain-wheel. 




Fig. 79. 



HIGH-SPEED CHAINS. 



57 



A geared chain formed of square open links made of malleable iron 
is now very extensively used in agricultural and other machinery requir- 
ing light geared chains. Fig. 79 A shows a side elevation of the chain 
and sprocket, and at B the form of the link and the method of fastening 
are shown. The sides and one end of each rectangular link are round in 
section, and the remaining end carries a hook-shaped projection which 
is as wide as the rectangular opening, and hooks over the roundad end of 
the adjacent link; the rectangular spaces between the hooks and oppo- 
site ends of the links receive the teeth of the wheel, which act against 
the rounded inner parts of the hooks as on round pins. The links 
are so made that they can be slid together sidewise when their planes 
are about at right angles. 

Very large chain-wheels may have cylindrical faces provided with 
suitably formed projections or hollows. 

The chief objections to chain gearing are the liability of stretching 
of the links forming the chain, and the wearing of the pins, or links, 
both of which tend to alter the pitch, and thus cause bad fitting between 
the chain and the wheel. The necessity for slow motion is also some- 
times a disadvantage. 

80. High-speed Chains. — None of the above-mentioned chains, even 
if very accurately made, can be run at high speeds without noise which 
rapidly increases with the wear. A form of high-speed chain has been 
developed by Hans Reynold known as the Reynold silent chain. It 
consists of links C of a peculiar form with straight bearing edges a, b, 
Fig. 80, which run over cut sprocket-wheels with straight-sided teeth 




Fig. 80. 
whose angles vary with the diameter of the wheel. The chain may be 
made any convenient width, the pins binding the whole together. One 
sprocket of each pair is supplied with flanges to retain the chain in place. 



58 CONNECTION BY BANDS OR WRAPPING CONNECTORS 

The upper figure shows a new chain in position on its sprocket, the bear- 
ing points of the links a,b,c being on the straight edges of the links only, 
not on the tops or roots of the teeth. The chain thus adjusts itself to 
the sprocket at a diameter corresponding with its pitch, and as any tooth 
comes into or out of gear there is neither slipping nor noise. The lower 
figure shows the position taken by a worn chain of increased pitch on 
the same wheel, a x and b t being the new bearing surfaces on the wheel W t . 
Morse Rocker-joint Chain. — This chain is an improved form of the 
silent-running type and is now extensively used in place of belting and 
gearing at speeds up to 2000 ft. per minute. In it the links are rounded 
at one end and made to fit the sprocket-wheel at the other, as shown 
in Fig. 81. The teeth of the sprocket-wheel are unsymmetrical, their 




Fig. 81. 

working faces d being more nearly radial, thus reducing any tendency 
to slipping in the chain. The ordinary pin bearings are here replaced 
by rocker-joints consisting of two pieces of hardened tool-steel, a and b, 
so shaped and arranged that in passing on and off the sprocket one 
piece rocks upon the other. Each link A has fast in one end the seat- 
piece a and at the other a rocker-piece like b, and is so shaped as to be free 
to move through the required angle on parts similarly held in the adjoin- 
ing links, such as b in link B. Each outside link is bent laterally so' 
that the large end comes under the small end of the adjoining link to 
permit of the proper engagement with both seat and rocker piece. The 
shouldered ends of the seat-pieces a are softened to allow their being 
riveted either to the outside links or to washers. 

To prevent undue vibration under high speeds and consequent wear, 
the rocker-pieces are so shaped that the contact surfaces are greatly 
increased when the chain is drawn straight, as shown at c, thus giving 
a broad bearing surface which, while stiffening the chain, also reduces 
the pressure on the parts designed for rolling except when the chain 
is passing over its sprockets. This chain, having so very little friction, 
has an exceedingly high efficiency. 

To keep the chain in place on its sprockets special guiding links are 
supplied which project below the chain into grooves turned in the 
sprockets. 



CHAPTER VI. 



LEVERS.— CAMS. 



81. Levers. — We often have occasion to transfer some small motion 
from one line to another; we will consider three cases depending on the 
relative positions of the lines of motion : 

1° Parallel lines. 

2° Intersecting lines. 

3° Lines neither parallel nor intersecting. 
In 1° we have the common lever (Fig. 82) , where, when the lever 
has a small angular motion, the points a and b receive motions propor- 
tional to their distances from the centre c, and — ( eg--- 
approximately in the parallel lines aa x and bb v 
If we suppose the lever to move, we have 

l.v. a _ac 



l.v. b 



Z=S 



I©- 



«i 



&i 



If P and W denote the forces applied at a 
and b respectively, we have, if they are in equilibrium, 

P be 



Fig. 82. 



PXac = WXbc, or 



W 



ac 



or the forces are inversely proportional to the lengths of their lever-arms. 

82. Bell-crank Lever. — In 2° we have the bell-crank lever ocb 
(Fig. 83), where a and b move approximately in the lines ad and bd, 

intersecting at d. Here, as in 1°, the 
velocities are proportional to the lengths 
of the lever-arms, or are proportional to 
the sines of the angles adc and cdb 
made by the line cd with the directions 
of the motions. 

Suppose the angle adb to be given 

and a bell-crank lever is required that 

will give a motion along ad equal to 

one-half that along bd. 

Draw the line dc, dividing the angle adb into two parts whose sines 

are directly proportional to the velocities of a and b. This may be done 

by erecting perpendiculars on ad and bd in the ratio of the required 




Fig. 83. 



60 



LEVERS.— CAMS. 



motions along those lines, and drawing through their extremities lines 
parallel to ad and bd respectively; the intersection of these lines at e 
determines the line de. Choose any point c in de, and drop the perpen- 
diculars ca and cb on ad. and bd respectively; then acb is the bell-crank 
lever required. 

Here, as in the previous case, if we suppose the lever to move, we 
have 

l.v. a, ac sin cda 
l.v. b be sin cdb' 
It is evident that, for a small angular motion, the movements in ad 
and bd are very nearly rectilinear, and they will become more and more 
so the farther we remove c from the point d. 

Any slight motion that may occur perpendicular to the lines ad and 
bd may be provided for by the connectors used. It is to be noticed, 
however, that for a given motion on the lines ad and bd these perpen- 
dicular movements, or deviations, are less when the lever-arms vibrate 
equal angles each side of their positions when perpendicular to the lines 
of motion, and they should always be arranged to so vibrate. Ey mov- 
ing the point c nearer to d, at the same time keeping the lever- 
arms the same, this perpendicular devia- 
tion could be disposed equally on each 
side of the lines of motion, which is 
advisable especially in cases where the 
deviation is allowed for by the spring of 
the connecting piece. This is shown in 
position a^!, Fig. 83. In Fig. 83 it will 
be seen that a and b move in opposite 
directions, while Fig. 84 shows the result 
if a and b are to move in the same direction. 
In 3° we have the lines of motion ad and be (Fig. 85) neither 
parallel nor intersecting. Draw ee 1} the common perpendicular to the 
lines ad and be; through e 1 draw 
e^ parallel to ed; construct a 
bell-crank lever ajjb, for the de- 
sired movements considered as trans- 
ferred to the lines be and e 1 c? 1 in 
one plane; draw ff t parallel to ee 1 
and equal to it, and further make 
af parallel and eqra 1 to 1 f l . The 
piece afffi will be the lever required. 
What has been done is this: a 
bell-crank lever has been con- 
structed in a plane, passing through FlG - 85 - 
the first line of motion and parallel to the second line, the transferred 




Fig. £4. 




CAMS AND WIPERS. 



61 



second line of motion in this plane being a projection of the second line of 
motion ; or the lever aJJ) has been arranged to transfer the motion from 
be to a x d x ; then the motion along a x d x has been transferred to the second 
line ad parallel to a x d x by moving the lever aj x parallel to itself an 
amount equal to the perpendicular distance between the two lines of 
motion, and connecting /, and / by means of a shaft, so that af and bf t 
turn together about jf v The rocker-arm on a locomotive, which trans- 
fers the motion from the link, inside of the engine frame, to the valve- 
rod, outside of the engine frame, is an example of this form of lever. 

83. Cams and Wipers. — A cam is a curved plate or groove which 
communicates motion to another piece by the action of its curved edge. 
This motion may be transmitted by sliding contact; but where there is 
much force to be transmitted, it is often by rolling contact. 

If the action of the piece is intermittent, it is sometimes called a 
wiper; that is, a cam, in most places, is continuous in its action, while 
a wiper is always intermittent: but a wiper is often called a cam not- 
withstanding. 

In most cases which occur in practice the condition to be fulfilled in 
designing a cam does not directly involve the velocity ratio, but assigns 
a certain series of definite positions which the follower is to assume 
while the driver occupies a corresponding series of definite positions. 

The relations between the successive positions of the driver and fol- 
lower in a cam motion may always be represented by means of a diagram 




Fig. 86. 

such as is given in Fig. 86, where the line Oabc represents the motion 
given by the cam. The perpendicular distance of any point in the line 



62 LEVERS.— CAMS. 

from the axis OY represents the angular motion of the driver, while the 
perpendicular distance of the point from OX represents the correspond- 
ing movement of the follower, from some point considered as a starting- 
point. Thus the line of motion Oabc indicates that from the position 
to 4 of the driver, the follower had no motion; from the position 4 to 
12 of the driver, the follower had a uniform upward motion 612; and 
from position 12 to 16 of the driver, the follower had a uniform down- 
ward motion 612, thus bringing it again to its starting-point. 

84. Where the cam acts upon the point for which the motion is 
given, as d in Fig. 86, and where the motion of the point is in a straight 
line, passing through the axis of the cam, it is only necessary in con- 
structing the pitch line of the cam to make the radii of such length 
as to bring the follower to the desired positions after the required inter- 
vals of rotation of the cam. Thus in Fig. 86 the point d, the axis of the 
roller, is to be still for one quarter of a turn; then it is to move up the 
distance de uniformly in one half a turn; and then down the same dis- 
tance uniformly in one quarter of a turn. Let the cam turn right- 
handed. Draw radial lines from the axis of the cam, at intervals cor- 
responding with those taken on OX in the diagram, in this case six- 
teen lines, at equal intervals. The pitch line from to 4 is an arc of 
a circle subtending 90° with a radius equal to od. In the next 180° 
there are eight equal intervals and the radii 06, 08, etc., are made equal 
to the distances 06, 08, etc., on the line of motion ode. The remaining 
90° is divided into four equal intervals, and the radii ol3, ol4, etc., are 
made equal to the distances ol3, ol4, etc., on ode. The curve through 
the points 4, 6, 8, etc., will give the pitch line of the cam, which line 
would cause the axis of the roller to move with the desired motion. 

If a roller is used, the cam outline lies inside the dotted line, as shown 
in Fig. 86, a distance equal to the radius of the roller, and may be found 
by striking arcs from the pitch line with a radius equal to that of the 
roller, as shown, and drawing a smooth curve tangent to these arcs. 
The heavy curve shows the cam outline, which, acting upon the roller, 
will give its axis the desired motion. 

It will be noticed that at the point 12, using the roller, the axis is not 
moved quite so far as e. The use of a roller will often be found to pre- 
vent the exact equivalent of the motion as given by the pitch line, the 
two normal curves overlapping at their juncture, thus cutting off a part 
of each. 

It will be noticed (Fig. 86) that the cam can only drive the roller 
in one direction, viz., away from the centre 0; in order to provide 
for the return, a spring or the weight would have to be relied upon, 
tending to force d toward 0. Now if we suppose the cam-plate to be 
extended beyond the roller, and cut a groove in it which would be the 



DIAGRAMS FOR CAMS GIVING RAPID MOVEMENTS. 



63 



envelope of the successive positions of the roller, by the principle of 
Fig. 14 it would act upon the roller equally well in either direction. The 
groove in this case would have parallel sides, and should be a little 
wider than the roller to prevent binding, the play being allowed by 
taking material from the non-acting surface of the cam. Cams are 
usually supplied with rollers, as they greatly reduce the friction and 



v 




85. Diagrams for Cams giving Rapid Movements. — It is very often 
the case that a cam is required to give a definite motion in a short inter- 
val of time, the nature of the motion not being fixed. We will now 
discuss the form of the diagram for such a motion. 

In the diagram shown in connection with Fig. 86 the follower had 
two uniform motions, and if the cam be made to revolve quickly, quite 
a shock will occur at each of the points where the motion changes, as a, b, 
and c; to obviate this the form of the diagram can be changed, pro- 
vided it is allowable to change the nature of the motion. 

Suppose we wish a cam to rapidly raise a body from e to / (Fig. 87), 
the nature of the motion to be 
such that the shock shall be 
as light as possible. 

If we draw the straight 
line Oa, we have the case of a 
uniform motion (as in Fig. 86), 
the body being raised from e 
to / in an interval proportional 
to ob ; here the motion changes 
suddenly at and a accom- 
panied with a perceptible 
shock. The line Ocda would be an improvement, the follower not 
requiring so great an impulse at the start or near the end of the motion, 
each being much more gradual than before. 

The body may be made to move with a simple harmonic motion, 
the diagram for which would be drawn as follows (Fig. 88). 

Draw the semicircle ehj on ef as a diameter; divide the time line 
Oh into a convenient number of equal parts (in this case ten), and then 
divide the semicircle into the same number of equal parts; through the 
divisions of the semicircle draw horizontal lines intersecting the vertical 
lines drawn through the corresponding points of division of the time 
line Oh, thus obtaining points, as a, b, c, etc. A smooth curve drawn 
through these points gives the full curve Oabcd . . . n. Here the body 
or follower receives a velocity increasing from zero at the start to a 
maximum at the middle of its path, when it is again gradually diminished 
to zero at /, the end of its path. 

This form of diagram gives very good results, and is satisfactory 
in many of its practical applications. 

















d 




a 


^2rff 


















b 


1 1 i 


J 


4 


I 

F 


IG. 


87. 


s 


i 


1 


X 



64 



LEVERS.— CAMS. 



A body dropped from the hand has no initial velocity at the start, but 
has a uniformly increasing velocity, under the action of gravity, until it 



n f v 




1«' 2 3 4 5 C 7 



9 10 1^(3 X 



Fig. 88. 



reaches the ground; similarly, if the body is thrown upward with the 
velocity it had on striking the ground, it will come to rest at a height 
equal to that from which it was dropped, and its upward motion is the 
reverse of the downward one, that is, a uniformly retarded motion. 

In designing a cam for rapid movement the motion of the follower 
should obey the same law of gravity, and have a uniformly accelerated 
motion until the middle of its path is reached, then a uniformly retarded 
motion to the end of its path. 

A body free to fall descends through spaces, during successive units 
of time, proportional to the odd numbers 1, 3, 5, 7, 9, etc., and the 
total space passed over equals the sum of these spaces. 

To develop a line of action according to this law upon the same time 

line Oh, and with the same motion 
ef, as before, proceed as follows : 

Divide the time line Oh into 

any even number of equal parts, as 

ten; then divide the line of motion 

ef into successive spaces propor- 

\ tional to the numbers 1, 3, 5, 7, 9, 9, 

\ 7, 5, 3, 1, and draw horizontal lines 

1 through the ends of these spaces, 

; obtaining the intersections a', b' ' , c' ', 

l etc., with the vertical lines through 

/ the corresponding time divisions 1, 

2,3, etc. ; a smooth curve, shown 

dotted in the figure, drawn through 

these points, will give the cam 

diagram. 

86. Heart Cam. — If the desired 
motion is to be on a line passing 
through the axis of the cam, uniformly up in one half a turn of 




Fig. 89. 



HEART CAM. 



65 



the cam and uniformly down in the remaining half-turn, a heart- 
shaped cam will be the result, as shown in Fig. 89. The curve for the 
pitch line of such a cam will be found to be the spiral of Archimedes, 
as it fulfils the polar equation for that curve, 

r = d-\-md, 

where r represents the radiant, d the distance of the starting-point c on 
the initial line cc t from the origin of co-ordinates o, 6 the angular 
motion of the radius vector, and md the increase in the radius vector 
for the angular motion d, or the motion of the cam. If the spiral starts 
at the origin o, d = 0, and the equation then becomes 

r = mQ. 

87. If the line of motion of the follower-point does not pass 
through the axis of the cam, the construction shown in Fig. 90 is used. 

Here ab is the line of motion of the follower, and the point a is to 
be uniformly raised through 
the distance ab while the 
cam turns uniformly left- 
handed five eighths of a turn; 
it is then to be suddenly 
dropped to its first position a, 
and remain there for the re- 
maining part of the cam rota- 
tion. Draw the small circle 
cc x c & tangent to the line cab 
with as a centre; all the 
positions of the line ab will be 
tangent to the small circle if 
we suppose the line ab to re- 
volve about right-handed; 
draw a circle through any 
point on the line of motion ab, as d, with as a centre, and make the 
angle dod 6 = coc Q . Divide the arc dd e into any convenient number of 
equal parts, and divide the distance ab into the same number of equal parts. 
From the divisions on the arc dd 6 draw lines d x c x , d 2 c 2 , etc., tangent to 
the circle cc x c Q . These lines, on rotating the cam, will coincide success- 
ively with the line of motion ab. To find a point on the cam outline 
.as e v draw an arc through a x with as a centre, and note e x where it 
intersects d x c x ) other points can be found in a similar way. The part 
of the cam producing no motion is made circular. 




Fig. 90. 



88. Involute Cam.— If the distance ab (Fig. 90) is equal to the 
arc cc t c 6 of the small circle, the point a would have a motion such as 
would be derived by unwrapping a band from the small circle, and the 



66 LEVERS.— CAMS. 

curve of the cam would then be the involute of the small circle, thus 
giving an involute cam. 

If the distance through which the point is to be raised and the cor- 
responding angular motion 6 of the cam are known, we have for the 
unknown radius r of the involute circle the equation 

n .~. , Distance 
rt) = Distance; or, r = ^ — — , 

6 being expressed in circular measure. 

After determining the radius r of the small circle, the line of motion 
can be located, and the cam constructed, as in Fig. 90. 

This form of cam is often used to raise the stamps of an ore-crusher, 
or for giving a uniform upward movement to a rod passing by the shaft 
of the cam. 

89. General Case. — Suppose we have given (Fig. 91) the position of 
the axis of a plate cam; that the cam turns uniformly right-handed 
and gives motion to the slide D, in a straight line ef by means of the 
lever deb centred at c, and the rod ab, the cam to act on the point d T 
on which as an axis a roller could be placed; that the slide D shall 
remain stationary for the first quarter of a turn, move uniformly up 
on ef an amount aa Q in the next half-turn, and then move with simple 
harmonic motion down an amount a 6 a in the remaining quarter of a 
turn, to find the pitch line of the cam. 

First, draw the motion diagram, where Oh is taken to represent 
360° of motion of the cam. For the first quarter of a turn there is no 
motion of D, thus giving the line &0 = l Oh, coincident with OX; 
for the next half -turn, from to 6 equal to %Oh, there is a uniform 
upward motion g6 = aa Q ; this would be indicated by the straight line Qg r 
and any intermediate position of D between a and a 6 could be found by 
drawing an ordinate at the cam position; similarly, for the next quarter 
of a turn the curve gh would represent the motion of Z), this curve 
being found by drawing the semicircle, shown on OY, with a diam- 
eter equal to aa G , the distance to be moved through, and then proceed- 
ing as in Fig. 88. Transfer the ordinates thus found, at the points of 
division of Oh, to the line of motion aa 6 as indicated. 

From the positions a, 1, 2, 3, etc., of the point a we can, knowing 
the length of the rod ab, and that the points b and d move in arcs of 
circles about c, determine the corresponding positions of the point d 
on which the cam is to act, marked d, 1, 2, 3, etc. 

Now, having found a series of definite positions of the point d corre- 
sponding with a definite series of positions of the cam. we will concern 
ourselves only with its motion on an arc about the centre c, relative to 
the cam turning on the axis 0. To find the pitch line of the cam, note 
that the relative motion of the cam and the point d moving about the 



GENERAL CASE. 



67 



centre c is the same whether we consider the point c as fixed and the 
cam to turn R.H. or whether we consider the cam as fixed and the lever 




Fig. 91. 

to revolve around the axis of the cam L.H., the point d at the same 
time having its proper rotation about c, as determined by the positions 
d, 1, 2, etc. Draw a circle through c with o as a centre, and divide it, 
proceeding in a left-handed direction, into parts corresponding with the 
divisions of the diagram. These points c , c 1} c 2 , etc., would successively 
pass through c. When c t is at c the point d t would be at the position 1 
on the arc through which the end d, of the lever bed, moves, d x being 
located at a distance d 1 c 1 = dc from c 1} and at a distance o\ from o equal 
to the desired distance of the point d from the centre of the cam. ' In 
the same way the points d 2 , d 3J etc., are found. Thus when the point 
c 6 is at c, d 6 will be at the point 6 on the arc through which the point d 
moves, d 6 being found by making c b d 6 = cd, and od 6 = o6, the distance 
from o to the desired position of d. A smooth curve through the points 
d , d v d 2 , etc., will be the pitch line of the cam. If greater accuracy is 
required, intermediate points should be found, or shorter intervals of 
rotation should be taken in those parts of the rotation where greater 
accuracy is desired. 

It is interesting to notice in connection with Fig. 91 that, if the 
edge of a thin plate is shaped like the diagram, and the plate is then 



6S 



LEVERS.— CAMS. 



moved uniformly a distance Oh along OX (its edge acting on a), while 
the cam disc turns uniformly once right-handed, a pencil carried at d 
would trace the outline of the cam on the disc. The diagram could also 
be drawn upon a piece of paper moving uniformly along OX a distance 
Oh, by a pencil carried at the point a, and moving under the influence 
of a uniform right-handed rotation of the cam. 



90. Positive-motion Cam. — When a cam actuates its follower equally 
well in both directions without external aid, as force-closure, it is called 
a positive-motion cam, the elements being so paired that only one motion 
is possible between them. 

There are other means than the use of a groove for insuring the 
positive motion of the follower not open to the objection of binding the 
follower roller in its groove. Two rollers might be used, working on 
opposite sides of the same cam, and always situated diametrically oppo- 
site each other; in such a case, the original cam outline, that is, the out- 
line passing through the centre of the rollers, must 'have a constant 
diameter equal to the distance between the centres of the rollers. 

Such a cam is shown in Fig. 92, where the rollers A and B bear on 

opposite sides of the cam, and are 
carried by the frame CC. During 
the first \ turn of the cam in the 
direction of the arrow, the roller 
A moves to A x with a uniform 
motion; it rolls on the cam sur- 
face from s to s 1} while B rolJs 
from s 2 to s 3 ; during the second 
J turn A remains at A i; the cir- 
cular part of the cam s^ acting 
upon it, s 3 s acting at the same 
time on B ; during the third \ turn A ± moves back to A under the action 
of s 2 s 3 , ss 1 acting on B; and during the last J turn^. remains stationary, 
s 3 s acting upon it, while s^ acts on B. 

Two cams might be arranged side by side 
on the same shaft, and act on two connected 
rollers, one running on one, and the other on the 
other cam; this would render possible a more ^ % 
complicated motion than that shown in Fig. 92. 
Two cams might also be used, as in Fig. 93, 
turning on separate shafts, but uniformly in 
relation to each other. 

Here the cams are counterparts of each 
other, and have the sums of the radii from the 
centres of A and B to the centres of the rollers a constant. This arrange- 




Fig. 92. 




Fig. 93. 




POSITIVE-MOTION CAM. 69 

ment has been used to operate the harnesses of looms, by connecting 
the end D of the lever CD to the harness frame. 

91. It is often the case that only a few positions of the follower of 
a cam are fixed, and it is not particular what the intermediate motion 
is. In such a case, the outline of the cam may be constructed by pass- 
ing arcs of circles through the fixed points of the cam in such a way as 
to make a smooth curve. 

Fig. 94 shows such a cam as applied to the lever of a punching- 
machine. Here the cam turns right-handed; the left-hand end of the 
lever is raised by the action of the arc 
of the cam extending from d to c, thus 
bringing the punch, the slider of which 
is worked by the end of the lever to 
the right of the fulcrum e, down to the 
metal. A wheel is placed at c to les- FlG - 94 - 

sen the friction during the punching ; after the punching, the lever is 
lowered by the action of the arc cb. The part of the cam dab is made 
circular about the axis 0, and for nearly one half a turn of the cam 
the lever remains at rest with the punch raised from the metal, thus 
allowing the workman time to adjust the plate for the next punching. 
The cam is often arranged to slide, by means of a key and key way, on 
its shaft, and is brought under the lever, when required, by means of a 
treadle moved by the foot of the person operating the machine. 

92. All cams thus far discussed have completed their action in one 
turn; by suitably shaping the follower, this action can be extended 
so as to require more than one turn, but the cams then become com- 
plicated, and are not much used on that account. 

Fig. 95 shows a cam where two turns are necessary to complete 

its action. The follower has the elongated form shown at F in order 

that it may properly pass the inter- 
section of the cam groove, which is 
the envelope of the follower. If the 
cam turns right-handed, there is a 
period of rest for the follower during 
the first \ turn of the cam, due to the 
FlG - 95 - circular groove ab; during the second 

i turn the lever moves to the dotted position under the action of be; 

cd then retains it for the third \ turn, and da returns it to. its first position 

during the remaining \ turn. 

93. Two cams might be made, by means of a system of levers and 
rods, to act on the same point, the motions governed by each cam being 
in lines situated in one plane and making right angles with each other; 
the point so governed could be made to trace any plane curve within the 
limits of its movements. 




70 



LEVERS.— CAMS. 




Fig. 96. 



If a plate cam is required to produce more than one double oscilla- 
tion of a vibrating lever during one revolution, its edge would be formed 
into a corresponding number of waves; if a groove is used, the centre 
line of the groove would have the same series of waves. 

94. Cylindrical Cams. — A cam may be made by cutting the proper 
groove around a cylinder; the motion of the follower would then be in a 

direction parallel to the axis of the 
cam. Such a cam, shown in Fig. 96 
A, produces a motion similar to that 
given by the cam shown at B. Both 
forms are very extensively applied to 
actuate the feed mechanisms in some 
machine-tools; their period of action 
only occupies a small part of their 
rotation, and comes just before the 
tool is ready for a cut. 
A cam like that shown in Fig. 96 A can be constructed by lay- 
ing out, upon paper, its motion diagram (Fig. 96 C) on a line ab, equal 
in length to the circumference of the cam cylinder, and wrapping this 
diagram around the cylinder, taking care that the line ab is kept in a 
plane perpendicular to the axis of the cylinder. The centre line of 
the groove being thus determined, the groove can be made the envelope 
of the follower when it moves along this centre line. 

A groove having a centre line acdeb (Fig. 96 C) might be cut in a 
flat plate which has a rectilinear motion along ab ; a follower moving on 
a line perpendicular to ab, supplied with a roller working in the groove, 
would have a motion very nearly the same as that obtained by A and B. 
If the lines of motion in A and B were straight instead of curved, the 
roller in C would have exactly the same motion as in A and B. Cam 
grooves, cut in plates having a rectilinear motion, are often used' in 
practice. 

A cylindrical cam having a helical groove is shown in Fig. 97. The 
follower F is made to fit the groove sidewise, and is arranged to turn 
in the sliding rod, to which it 
gives motion in a line parallel 
with the axis of the cam. The 
guides for this rod are attached 
to the bearings of the cam, A 
and B, which form a part of the 
frame of the machine. A plan 
of the follower is shown at G: 
its elongated shape is necessary 

so that it may properly cross the junctures of the groove. In this cam 
there is a period of rest during one-half a turn of the cam at each end 




CYLINDRICAL CAMS. 



71 






Fig. 98. 



of the motion; the motion from one limit to the other is uniform, and 
•consumes one and one-half uniform turns of the cam. 

The cylinder (Fig. 97) may be increased in length, and the groove 
may be made of any desirable pitch; the period of rest can be reduced 
to zero, or increased to nearly one turn of the cam. A cylindrical cam, 
having a right- and a left-handed groove, is often used to produce a 
uniform reciprocating motion, the right- and left-handed threads or 
grooves passing into each other at the ends of the motion, so that there 
is no period of rest. 

The period of rest in a cylindrical cam, like that shown in Fig. 97, 
can be prolonged through nearly two turns of the cylinder by means 
•of the device shown in Fig. 98. A 
switch is placed at the junction of 
the right- and left-handed grooves 
with the circular groove, and it is 
provided that the switch shall be 
capable of turning a little in either 
direction upon its supporting pin, 
while the pin is capable of a slight 
longitudinal movement parallel with 
the axis of the cylinder This 
supporting pin is constantly urged to the right by a spring, shown in A, 
which acts on a slide carrying the pin; when in this position the space a 
between the switch and the circular part of the groove is too small to 
allow the follower to pass, and when the follower is in the position shown 
in B, the spring is compressed; then, if the follower moves on, the 
space behind it is closed, as the spring will tend to push the support 
to the right, and swing the switch on the follower as a fulcrum. 

If the cam turns in the direction of the arrow, in A the shuttle- 
shaped follower is entering the circular portion of the groove, and leaves 
the switch in a position which will guide the follower into the circular 
groove when it again reaches the switch; in B the switch is pressed 
toward the left to allow the follower to pass. As motion continues, the 
support of the switch is pressed to the right, and the switch is thrown 
into the position shown in C ready to guide the shuttle into the return- 
ing groove. The period of rest in this case continues for about one and 
two-thirds turns of the cylinder. 

95. Fig. 99. shows a method of drawing a cylindrical cam by means 
of the projection of its pitch line. Let the point p move uniformly 
to the right the distance pp t in 1^ turns of the cylinder in the direction 
indicated; then let p remain stationary for a quarter-turn of the cylinder; 
then move to the left the distance p x p with uniformly accelerated and 
retarded motion in one turn; and finally remain still for a half-turn. 
Intervals of rotation may be chosen as may be desirable. In the figure 
the surface of the cylinder is divided into spaces each subtending 



72 



LEVERS.— CAMS. 




'ig. 99. 



45°. For the uniform motion to the right the line of motion parallel 

to the axis of the cylinder and 
equal to pp x is divided into ten 
equal parts, since there are ten 
45° intervals in 1^ turns. A 
smooth curve drawn through the 
points a v b v c v etc., the intersec- 
tions of perpendiculars from the 
points a, b, c, etc., on the line of 
motion with the corresponding 
elements on the surface of the 
cylinder, will give the pitch line of 
the cam. For the motion to the 
left the line of motion is divided into 
eight spaces (since there are eight 45°" 
intervals in one turn) , and in propor- 
tion to the numbers 1, 3, 5, 7, 7, 5, 3, 1, 
and the intersections of perpendicu- 
lars from the ends of these spaces 
with the corresponding elements will 
give the points through which the 
pitch line is drawn. The pitch line for 
the two periods of rest will be parallel 
to the base of the cylinder as shown. 
Fig. 100 is a development of Fig. 
99, and could be plotted in a similar 
manner to that described for Fig. 99, 
as is indicated on the figure, except- 
ing that the elements are spaced on the development of the cylinder , 
in this case becoming eight equal spaces. Wrapping this development 
around the cylinder, as was suggested in connection with Fig. 96, would 
give the pitch line as projected in Fig. 99. 

If the path of the follower, in Fig. 97, is inclined to the axis of the 
cam, the groove would be cut in a cone, or hyperboloid, generated by 
revolving the line of action about the axis of the cam. 

96. A conical cam might also be constructed to actuate a follower in 
a line perpendicular to its axis, and, by changing the position of the base 
of the cone relatively to the follower (the cam sliding along its axis), a 
variation in the motion could be obtained. A conical or cylindrical 
grooved cam might be made to actuate a roller in two directions radially 
and axially, such roller being supported by an arm moving on a spherical 
joint or its equivalent. A plate cam might also be arranged to turn 
slightly relative to its shaft, by means of a helically grooved sleeve and 
roller, and the relative times of the motions would thus be changed, 
an arrangement made use of in some governing devices. 




Fig. 100. 




CHAPTER VII. 
LINKWORK. 

97. Linkage. — Four pairs of elements may be combined, and this 
combination may take place in different ways. Suppose we have given 
the four parallel cylindric pairs, Fig. 101, each element of one pair rigidly 
joined to one element of another pair, we shall have an endless chain, 
or linkage, returning on itself. 

Link. — The rigid body joining two elements of different pairs is 
called a link, a linkage being made up of a number of links. Fig. 101 
may be taken as an example of a simple 
linkage, which consists of four pairs, each 
being a cylindrical pin fitting a correspond- 
ing eye, the axes of the pins being parallel. 
Here each link has a motion in a circle 
relative to its adjacent link; but every 

motion of any link must, according to its connection, be accompanied 
by alterations in the positions of the remaining links. Hence, if we 
consider one link of the chain to be fixed, the motions of the remaining 
links may be referred to it, and their relative motions determined. 

When in such a closed linkage we consider one of the links as fixed, 
we have a mechanism. Any link may be fixed, thus giving rise to as 
many mechanisms as the linkage has links. The fixed link forms a part 
of the frame of the machine, and may have a peculiar form. 

The different parts of a linkage are named according to the motion 
they have in respect to the stationary link or frame. 

Cranks, Levers, and Beams. — Links which turn or oscillate are called 
cranks, levers, or beams; the term crank being usually applied to links 
making complete turns, as the crank of a steam engine. 

Connecting-rod ; Coupling-rod.— The rigid link connecting the oscil- 
lating or rotating links is called by various names, as connecting-rod, 
crank-rod, eccentric-rod, coupling-rod, parallel-rod, etc., according to its 
location. It is attached to the pieces which it connects by pins or ball- 
and-socket joints, and maintains the distance between the centres of the 
pins or joints invariable. Hence its centre line is called the line of 
connection, and the centres of the pins are called the connected points. 

Crank-arm.— In a turning piece the perpendicular let fall from the 
turning point upon its axis of rotation is called the arm or crank-arm. 

73 



74 



LINKWORK. 



98. Angular Velocity Ratio. — Before proceeding to the discussion 
of the different mechanisms that can be derived from the simple linkage 
shown in Fig. 101 , we will determine the angular velocity ratios of the 
connected oscillating links. This may be found in two ways: 

1° By reference to the instantaneous axis of the connecting link. 
Let ABCD (Fig. 102) represent the linkage, D being the fixed link, 

A and C oscillating about the 
points a and d respectively. 
Then is the instantaneous axis 
about which B is revolving at 
the given instant. Produce be 
to meet ad at e, and draw as, dr, 
and ot perpendicular to be. By 
reference to the instantaneous 
axis, the l.v's of c and b, for 
the instant, are proportional to 




Fig. 102 
co and bo respectively, or 



l.v. c 



CO 

bo' 



l.v. b 

But the a.v. of C is equal to the l.v. of c divided by the radius cd, and 
the a.v. of A is equal to the l.v. of b divided by ab; so we may write, 
noticing that the triangles oct and dcr are similar, as are also obt and abs, 
l.v. c oc 
a.v. C cd cd 

ob cd 
ab 



a.v. A 



cd 
l.v. b 



__oc ab _ot as _as 
ob dr ot 



dr 



ab 



and, by the similar triangles ase and dre, 

a.v. C _as_ae 
dr de' 



(25) 



a.v. A 

2° By the resolution of velocities. 

In Fig. 102 let bb 1 represent the l.v. of b; bf will be the component 
of this l.v. along be, and cc t will therefore be the l.v. of c, cg = bf being 
the component of the l.v. of c along be; 

l.v. c_cc i 

'':' Lv76~&V 

and since a.v. equals l.v. divided by radius, and the triangles cc x g and 
dcr are similar, as are also bbj and abs, we may write 



cc, 



bb t eg as 



a.v. C _ 

a.v. A cd ' ab dr bf 
but the components eg and bf along be are equal 

a.v. C _as _ae 
= Jr = 



(26) 



a.v. A dr de 
Thus the angular velocities of the connected oscillating links are to 



L. V. RATIO OR A. V. RATIO IN LINKAGES. 



75 



each other inversely as the segments into which the centre line of the 
connecting link divides the line of centres. Or, if the intersection of 
the line of centres and the connecting link is at an inconvenient distance, 
as will often happen, the rule may be stated by using the first ratio in 
equation (26) : the a.v's of the connected oscillating links are to each 
other inversely as the perpendiculars from the centres of oscillation 
to the centre line of the connecting link. 

This a.v. ratio of course varies for every relative position of the 
links; but if the perpendicular from the instantaneous axis to the 
centre line of the connecting link should fall at the intersection of the 
centre line of the connecting link arid the line of centres, that is, in 
Fig. 102, if the points t and e should coincide, the a.v. ratio is essen- 
tially constant for slight movements in either direction. The same 
would be true should the points b and c be moving in lines parallel to 
each other. 

99. Diagrams for Representing Changes in the L. V. Ratio or A. V. 
Ratio in any Linkage. — To obtain a clear knowledge of the change in 
velocity ratio in any linkage a diagram may be drawn where the 
abscissae may represent successive positions of one of the oscillating 
links, and the ordinates represent the 
a.v. ratio of the oscillating links. A 
smooth curve through the points thus 
found would show clearly the fluctua- 
tions in the a.v. of one of the links 
relative to the other. A curve for l.v. 
ratio could be similarly plotted. 

In the linkage shown in Fig. 103 
let A turn uniformly L.H., to draw a 

curve to represent the ratio 



120>— t-c;.. 




for a complete rotation of A. 





/ 


<* 


\ 


















1 


/ 






\ 
























\ 
















1 
























30° 6 


0- ! 


0° 1 


K)°li 


0°1 


V 2 


0°2 


C 2 


C 3 


K>°3 


"1 










^tC 














ORDINATES = 
ABSCISSAE = P( 






A 










a ;t-, a 

)SlTIONS 


OP 



















































Fig. 104. 



the curve shown in Fig. 104. 



a.v. A 

Take positions of A at intervals of 30° 
and draw perpendiculars from d and 
a to be in each of its positions. The 
ratio of the two perpendiculars in 
each position will give the a.v. 
ratio: thus, starting with A as 

given in the figure, we have =0; 

a.v. A. 

in the position ab x c x d we have 

a.v. C as x 

7~ = T~>* e ^ c - .Plotting these 

a.v. JA clr x 

values as ordinates and the 30° 

positions of A as abscissae will give 



76 



LINKWORK. 




Fig. 105. 



ioo. Crank and Rocker. — Let the link D (Fig. 105) be fixed, and sup- 
pose the crank A to revolve while the lever C oscillates about its axis d. 

In order that this may occur, we 
must always have the conditions: 

r,A+B+C>D. 
2°, A + D+OB. 
3°,A + B-C<D. 
4°,B-A + C>D. 

1° and 2° must hold in order 
that any motion shall be possible; 
3° can be seen from the triangle ac 2 d, in the extreme right position ab 2 c 2 d, 
which must not become a straight line; and 4° can be seen from the 
triangle ac x d, in the left extreme position ab x c x d. 

There are two points c ± and c 2 in the path of c at which the motion of 
the lever is reversed, and it will be noticed that if the lever C is the 
driver, it cannot, unaided, drive the crank A, as a pull or a thrust on 
the rod B would only cause pressure on a, when c is at either c y or c 2 . 
If A is the driver, this is not the case. 

Dead Points. — The two points in the path of the crank at which it 
is impossible to start it by means of the connecting-rod alone are called 
dead-points. 

The above form of linkage is applied in the beam engine as shown 
in Fig. 106, the link D being formed by the engine frame; correspond- 
ing parts are lettered the same as in Fig. 105. 
The instantaneous axis is, for the position 
shown, at o, and for the instant the l.v. of b 
is to the l.v. of c as ob is to oc, or as bf is to ce, 
the line ef, drawn parallel to be, being made 
use of when the point o comes beyond the 
limits of the drawing. Fig. 106. 

The angle through which the lever C (Fig. 105) swings can.be calcu 
lated for known values of A, B, C, and D. 

From the triangle ac 2 d 

C 2 +D 2 -(B+A) 2 



r"""E 


d 


^T / 
V v 

a\ 


ii i . 



cos adc 7 
and from the triangle ae x d 
cos adc, 



2CD 

C 2 +D 2 -(B-A) 



2CD 
and c l dc 2 = adc 2 — adc x . 

Thus the two angles adc 2 and adc x can be calculated, and their differ- 
ence will give the angle required. 

If the link B is made stationary, the mechanism is similar, the 
only difference being in the relative lengths of the connecting-rod and 
stationary piece or frame. 



DRAG-LINK. 



77 




101. Drag-link. — If the link A is made the stationary piece or frame 
(Fig. 107), the links B and C rotate about a and b respectively, that is, 
become cranks, and D 
becomes a connecting- 
rod. This mechanism is 
known as the drag-link. 

In order that the 
cranks may make com- 
plete rotations, and 
that there may be no 
dead-points, the fol- 
lowing conditions must 
hold: 

1° Each crank must 
be longer than the line FlG - 1Q 7- 

of centres, which needs no explanation. 

2° The link cd must be greater than the lesser segment cj and less 
than the greater segment c 4 d 2 , into which the diameter of the greater 
of the two crank circles is divided by the smaller circle. This may be 
expressed as follows: -..;•, 

cd > A + B — C (see triangle ac 4 d 4 ) ; 
cd<B+C— A (see triangle bc 2 d 2 ). 
Producing the line of the connecting-rod until it intersects the line of 
centres at e, and dropping the perpendiculars as and br upon it, we have 

a.v. B _be_br 
a.v. C ae as' 

In the positions abc^ and abc 3 d 3 when cd is parallel to the line of 
centres, the angular velocities of B and C are equal, as the perpendicu- 
lars br and as then become equal. 

If C revolves left-handed and is considered the driver, it will be 

noticed that between the positions abc 3 d 3 
and abc^ the link B is gaining on C, 
and between abc^ and abc 3 d 3 it is fall- 
ing behind C. Therefore from abc 3 d 3 
to abc x d x we have, as e falls to the right 
of A. 




Fig. 108. 
while from abc^ to abc 3 d 3 , 



a.v. B br 

77 = — > unity , 

a.v. C as J ' 



as e would fall to the left of A, we should 
have the a.v. ratio less than unity. 

102. The Double Rocking Lever. — The remaining case is shown in 
Fig. 108, where C is the fixed link, and the levers B and D merely oscil- 



78 



LINKWORK. 



late on their centres c and d; the extreme positions are shown at a 1 6 1 




and a 2 b 2 . 



Fig. 109. 




Fig. 110. 



A modified form of this mechanism (Fig. 
109) is used in mechanisms for tracing straight 
lines, commonly known as "parallel mo- 
tions. " 

103. The following figures show a few ap- 
plications of the preceding forms of linkwork: 
Fig. 110 shows a case of the crank and rocker as applied in wool-comb- 
ing machinery. Here the crank ab turns uni- 
formly on its axis a, while cd oscillates about d; 
both axes a and d are attached to the frame of 
the machine, which forms the fixed link ad. 
The connecting-rod cb is prolonged beyond b, 
and carries a comb e at its extremity, which 
takes a tuft of wool from the comb / and trans- 
fers it to the comb g, both combs / and g being 
attached to the frame of the machine. The 
full lines show the position of the links when 
the comb e is in the act of rising through the 

wool on /, thus detaching it, and the dotted lines show the position of the 
links when the comb e x is about to deposit the tuft of wool on g. The 
same combination inverted is used in some forms of wool-washing 
machines. 

In Fig. Ill, which represents Morgan's feathering paddle-wheel, an 

application of the drag-link is 
shown. Each float is attached 
to one end of a link D, and turns 
on a bearing in the paddle-wheel 
frame carried by the paddle- 
shaft; the lines joining these 
bearings with the axis of the 
paddle-shaft give the links B. 
To the other end of D the links 
C, which revolve about a fixed 
centre in the paddle-box, are 
attached. As the space for the 
links C is limited, the arrange- 
ment shown in the figure is used 




Fig. 111. 



instead of having the links turn on one pin and located side by side. 
Here the link C is attached to a solid ring which rotates about the centre 
in the paddle-box, and the other links, corresponding to C, are attached 
to this ring, and have motions very near what they w r ould have if they 
turned about the axis of the ring. The links are lettered the same as 



PARALLEL-CRANKS. 



79 




Fig. 112. 



in Fig. 107, A being the fixed link, B and C the cranks, and D the con- 
necting-rod. 

If we suppose the paddle-wheel to have no slip, the proper angle for 
the floats to enter the water without disturbance can be found as follows: 
Take one of the floats, as shown just after entering the water, and, as 
its motion is made up of two com- 
ponents, one being due to the 
motion of the vessel while the other 
is due to the rotation of the wheel, 
draw the parallelogram of motions, 
and it will be found that the float 
while in the water should always 
have the line of its face pass through 
the highest point of the wheel. 

Another example of the drag-link 
mechanism is shown in Fig. 112, 
which represents a Lemielle ventilat- 
ing-machine. The apparatus con- 
sists of a circular chamber with closed 
ends, having a passage M for the 
inlet, and one N, on the other side, 
for the discharge of air. A revolv- 
ing drum B centred on the axis a carries a series of vanes cd free to 
turn on the axes d, and at the same time governed in their positions by 
the links be attached to their free ends, and turning about a fixed centre 
b in the end of the chamber. It can easily be seen that, when the drum 
revolves in the direction indicated by the arrow, air will be drawn through 
the passage M and delivered to N; the vanes, opening out on passing 
the inlet and closing on passing the outlet, sweep the air before them. 

104. Parallel- cranks. — If in Fig. 105 of the four-bar linkage, we 
make the opposite links equal, i.e., A = C and B = D, the crank-chain 

becomes a parallelogram, as ABCD 
in the lower part of Fig. 113. The 
lever C then becomes a crank equal 
to A, and (if D is fixed and some 
means of passing the dead-points is 
provided) moves through the same 
angle as A. The nature of this 
mechanism is not changed by fixing 
any. of the other links. 

If, in the combination of two 
equal cranks, A and C (Fig. 113) 
^ IG * with a connecting-rod B, equal in 

length to the distance between centres of cranks, the crank A rotates 




Ci 



-^ B 



Bi 



sp^ 



4" 



80 JLINKWORK. 

right-handed until it comes over D, thus bringing the four links in line 
with each other, it will be noticed that in this position (one of the dead- 
points) a further motion of A might cause C to turn either right-handed 
or left-handed. That is, for any given position of A, except the dead- 
points, there are two possible positions of C, which can be found by- 
drawing an arc of radius B about the extremity of A as a centre, and 
noting where it cuts the path of the extremity of C. Thus a uniform 
right-handed rotation of A might cause a uniform right-handed rota- 
tion of C, or a variable left-handed rotation, as shown in Fig. 116. To 
prevent this change of motion, and to insure the passage of the cranks 
by their dead-points, two sets of equal cranks may be combined, as 
shown in Fig. 113; the angle between the two sets of cranks being com- 
monly taken 90°, so that when one set of cranks is at a dead-point the 
other is in its best working position. Then a uniform rotation of the cranks 
A, A t will cause a uniform rotation of C, C v thus giving a uniform and con- 
tinuous velocity ratio between the axes of the two sets of cranks. Loco^ 
motives with coupled drivers are familiar examples of this arrangement. 
Fig. 114 shows another method of passing the dead-points where a 

third equal crank, C v is placed in the 
plane of the other two, and con- 
nected to them by links equal in 
length to the distance between its axis 
and the axes of the other cranks 
respectively. 

Two cranks at right angles to each 
other, and located in one plane, 
could be connected with two others 
also at right angles, and located in 
another plane parallel to the first, by means of two parallel con- 
necting-rods sufficiently offset to enable them to clear each other in 
their motion, the distance between the two crank planes being made 
sufficient to admit of such an arrangement. This arrangement is 
practically of little value, especially when much force is to be trans- 
mitted, as offset-rods, unless made very heavy, are likely to bend and 
cause binding on the crank-pins. 

It will be seen, on reference to Figs. 113 and 114, that the connecting- 
rods B, B ly move in such a way that they are parallel to the line con- 
necting the axes of the equal cranks which carry them, in all their posi- 
tions, and also that all points in these rods move in circular paths of a 
radius equal to the length of the crank, i.e., the rods may be said to 
have circular translation (§ 25). 

Parallel-rod. — The term parallel-rod or coupling-rod is used to 
designate the rods, such as B and B t (Fig. 113), employed to connect the 
driving axles of locomotives. 




ANTI-PARALLEL CRANKS. 



81 




Fig. 115. 



If two circular rings, cef and dgh (Fig. 115), are arranged to turn on 
their centres b and a respectively, with equal angular velocities, they 
may be connected by a series of equal 
parallel links, cd, eg, fh, etc., equal in 
length to the distance ab between the 
centres of the rings. These links would 
then have motions in circular paths, but 
would always remain parallel to ab, the 
link A being here fixed instead of D as 
in Fig. 113. This combination has been 
used in wire -rope machinery, where it 
is necessary that the wires of the rope 
be laid without having any twist put 
into them. As the rope in being drawn 
from the laying block does not turn, the individual wires also must not 
turn; this is accomplished by attaching the bobbins E, which carry the 
wires, to the parallel links cd, eg,, etc., which have circular translation 
but no rotation, the laying block turning with the circular ring dgh on 
an axis through a perpendicular to the plane of the paper. 

105. Anti-parallel Cranks. — It was stated in § 104 that, in the com- 
bination of two equal cranks with a connecting-rod, equal in length to 
the distance between centres of cranks, a right-handed rotation of one 
of the cranks might cause a left-handed rotation of the other. Fig. 116 
shows such a case where 'a uniform right-handed rotation of ab causes a 
variable left-handed rotation of cd, when some method of passing the 
dead-points is provided. Now if we can provide these links, which 
are capable of two motions at their dead-points, or change positions, 
with pairs of elements so formed that they will give the required mo- 
tion at these change points, the passage of the dead-points will be 
secured. 

In order to do this, we must, by the principles of § 27, determine the 

axoids or centroids of the bodies to be 

paired. These bodies are . here, for 

example, the two cranks A and C, or 

otherwise the two links B and D. In 

,>• the case of the general four-link chain, 

\ shown in Fig. 101, the centroids are very 

j complicated figures; here they are 

/ made very simple by the equality of 

/ the opposite links. (The linkage taken 

in § 27 to illustrate a centroid was 

this linkage.) Remembering that the 

cranks are always to revolve in opposite 

directions, the centroids will be found to have the forms shown in Fig. 116. 




Fig. 116. 



82 



LINKWORK. 



For the links ab and cd, the shorter pair, they are ellipses, having 
their foci at the ends, a, b and c, d, of the cranks, and their major axes 
equal in length to the links be and da. The instantaneous centre moves 
backwards and forwards along the links be and da, being always found at 
their intersection, as e. For be and da the centroids are hyperbolas, their 
transverse axes fg and hk lying on the links themselves and being equal 
to ab = cd; their foci are the points b, c and a, d. The instantaneous 
centre traverses each branch of the curve to infinity, turning from 
— oo along the other branches. Thus the two ellipses or the two hyper- 
bolas could replace the linkage. In § 58 it was found that two equal 
rolling ellipses were equivalent to this linkage, and also in § 61 the same 
was found to be true of two equal rolling hyperbolas. 

If it be required to pair the two opposite links at their change points, 

a pair of elements must be employed 
in each case; such pairs need not, 
however, go further than correspond 
to the elements of the rolling conies 
in contact at the change positions. 
If the links chosen be the two shorter 
ones, ab and cd, these are the ele- 
ments of the ellipses at the ex- 
tremities of their major axes. By 
putting a pin and a gab (or open 
eye) at these points, as shown in 
Fig. 117 at I and m, l t and m lf the 
I gearing with l l causes the passage 
of one dead-point, and m gearing with m x causes the passage of the 
other. 

If the two longer links are to be paired instead of the two shorter 
ones, we have only to notice that 
the two vertexes g and h, and / and 
k touch each other in the change 
positions. By placing at these points 
a pin and corresponding gab (Fig. 
118) we have again a pairing which 
effectually closes the chain. Thus 
we have two solutions of the prob- 
lem before us. If it were desired, 
it would be possible to close the chain at one dead-point by one method, 
and at the other by another. A few teeth on the ellipses or hyperbolas 
mj^ht be used in place of the pin and gab. 

/ v io6. This linkage replaced by its centroids, the rolling ellipses, is 
used to give a quick-return motion on some forms of slotting-machines. In 
Fig. 119 ; if ab turns uniformly, R.H., the upward motion of the slide 




Fig. 117. 



mechanism becomes a closed chain 




Fig. 118. 



ANTI-PARALLEL CRANKS. 



83 



E, operated by a crank de placed at an angle of 90° with cd, would 
begin in the position abed and end in the 
position ab&d, occupying a time proportional 
to the angle a; while the time occupied in 
the downward motion of E would be pro- 
portional to the angle /?, or 

advance of E : return of E=/3 : a. 

If the distance between the centres ad is 
given, and the ratio of advance to return, 
the angle bad is determined and the length 
of the crank ab ( = cd) can be readily calcu- 
lated; and from § 105 we know that a, b 
and c, d will be the foci of the ellipses and 
that ad ( = be) will be the length of the major 
axis of each. 

If the l.v. of the point b is known, the 
l.v. of the slide may be found for any posi- 
tion by the method in § 22, with the excep- 
tion of the two positions in which the links 
ab, be, and cd lie in the same straight line. 
To determine the l.v. at these two posi- 
tions it is necessary to refer to the rolling 
ellipses. In the position ab 2 c 2 d the ellipses 
would be in contact at the point f 2 g 2 and we 
should have l.v. of g 2 equal to the l.v. of 

the point f 2 in contact with it; therefore, if b 2 h represents the l.v. of 
b 2 , f 2 k will be the l.v. of the point / 2 and so of g 2 , and c 2 l will be the 

l.v. of c 2 . The l.v. of E for the position E 2 would be c 2 lx — 

cd 
The a.v. ratio of the cranks ab and cd may be found for any position 
by the law deduced in § 98, excepting the two positions referred to in 
discussing the l.v. In the position abed we have 

a.v. cd _am 
a.v. ab drn 

but in the position ab 2 c 2 d, by referring to the rolling ellipses, 




119. 



a.v. c 2 d _ af 2 < 
a.v. ab 2 _ dg 2 



that is, the a.v's are inversely as the radii of contact of the rolling sur- 
faces. 

To obtain a clear idea of the l.v. of the slide E as ab turns uniformly, 



84 



LINKWORK. 



a curve should be drawn having for ordinates the l.v. of the slide F, 
and for abscissas the corresponding angular positions of ab. 

107. Slow Motion by Linkwork. — The simple linkage shown in Fig. 

105 can, if properly proportioned, 
be made to produce a slow motion 
of one of the cranks. Such a com- 
bination is shown in Fig. 120, where 
two cranks A and C are arranged to 
-p 120 turn on fixed centres and are con- 

nected by the link B. If the crank 
A is turned right-handed, the crank C will also turn right-handed, but 




LfA 




Fig. 121. 



with decreasing velocity, which will 
become zero when the crank A 
reaches position A lf in line with the 
link B 1 : any further motion of A 
will cause the link C to return 
toward its first position, its motion 
being slow at first and then gradually 
increasing. This type of motion is 
used in the Corliss valve-gear, as 
shown in Fig. 121. The linkage 
abed, moving one of the exhaust- 
valves, will give to the crank cd a 
very slow motion, when c is near c 1} 
when the valve is closed, while between c and c 2 , when the valve is open- 
ing or closing, the motion is much faster. The same is true for the ad- 
mission-valves, as shown by the linkage aefg. 

108. Forces Transmitted by Linkwork. — If we know the force 
applied at some point in a linkage, as .the pull on the end of a lever, 

it is possible, by equating the mo- 
ments of the forces acting around 
each axis of rotation, to deter- 
mine the force resulting at some 
other point in the linkage, neglect- 
ing the losses due to friction. In 
the linkage in Fig. 122, let a force 
F act at the point a in the bell- 
crank lever abc and at right angles 
with ab; it will cause a pull F x 
at c in the direction of the link cd 
which will act on the point d of the lever def with the same intensity F t 
in the same direction. (The arrows in these figures represent the direc- 
tions in which the forces may be considered to act, but do not repre- 
sent the magnitudes of the forces unless otherwise stated.) Since the 




TOGGLE-JOINT. 



85 



two forces F and F t acting around the axis b are balanced, their moments 
about b must be equal; therefore we have 

FXab 



FXab = F t Xbg and F x = 



tg 



and the force W which would be exerted by the point / against a resist- 
ance acting at right angles with ef would be found by the relation 

F t Xeh=WXef. 



W 



F\Xeh_ 



„ ab eh 
FXj-X-j. 

bg ef 



If the links are proportioned as in the figure, so that for a uniform 
R.H. motion of abc the lever def has a decreasing motion, we should 
find that for a given resistance, W, at / the force F required at a would 
diminish, its limit being zero at the moment when be and cd come into 
line; or if we consider a constant force F at a, the force W which could 
be exerted by / would increase, its limit being infinity, which means 
that a very large force may be produced by such a linkage, when near 
this position of slowest motion, by the application of a relatively small 
force. This principle is used in many forms of riveting, shearing, and 
punching machinery. 

109. Toggle-joint. — There are two forms of this joint shown in 
Figs. 123 and 124. In each case the point a is fixed and the point c 




Fig. 123. 



Fig. 124. 



slides on the line ac. If a force F is applied at b in the direction indi- 
cated, its moment arm about a will be as, and the moment arm about a 
of the thrust F x along the connecting link be will be ar; therefore 

FXas = F 1 Xar and F.=F-. 

ar 

Using the parallelogram of forces at the point c as indicated, 

F t cos a — W. 

.'. W = FX--Xcosa. 
ar 

As the links ab and be come more and more into line, the distance 
ar becomes smaller, the component of the thrust along ac approaches 



86 



LTNKWORK. 



nearer and nearer to the thrust on be, and when the links are in line the 
thrust along be is theoretically infinite, ar being then equal to zero. 
Fig. 125 shows a metal-shearing machine, in which a slow motion 

and consequently an advantage in 
power is obtained by means of link- 
work. Here the long lever A is 
formed by a continuation of the 
crank ab; the crank or lever C turn- 
ing on d is connected to ab by the 
link be, a and d being fixed centres 
carried by the frame of the machine, 
which forms the fourth link of the 
chain. The metal to be sheared is 
placed at S, and the power is applied at the end of the lever A. The 
operation of the machine can be easily understood from the figure. 
The links ab and be, c moving nearly in a straight line, form a toggle-joint. 




Fig. 125. 



-If we combine the links dc, 



i io. Double Oscillation by Linkwork 

ce, and ie as shown in Fig. 
126, d and i being fixed cen- 
tres, with a crank ab (turn- 
ing on the fixed centre a), 
and a connecting-rod be, the 
lever ie can be made to make 
a double oscillation during 
one rotation of the crank 
ab. In order that this may 
properly occur, the extreme 
positions of the point c should 
be equidistant from the cen- 
tre line de, and the extreme 
positions of e should be 
equidistant from the per- 
pendicular dropped from the 
centre i on the line de. If 
we continue the link ie to /, and connect it with the links fg and gh, h 
being a fixed centre, and so chosen that the extreme positions of g are 
equidistant from the perpendicular dropped from h upon ig, the lever 
hm will make four oscillations for each rotation of the crank ab. 

in. Linkwork with One Sliding Pair. Line of Connection in a Slid- 
ing Pair.— Returning again to the four-bar linkage of Fig. 105, we can 
substitute for the link Ca small sector of an annular cylinder, and en- 
close this in a circular slot (Fig. 127) rigidly connected with the 
centre about which the crank A revolves. If the centre line of the 




Fig. 126. 



THE SLIDING-BLOCK LINKAGE. 



87' 





Fig. 128. 



slot be placed at a distance from d equal to the distance dc in Fig. 
105, and if the link B is at- 
tached in the centre of the 
sector, the link B has exactly 
the same relative motions as it 
would have had had it been con- 
nected to the link C. The ele- 
mentary link D and the sliding 
block C thus take the place of the links D and C of Fig. 105. 

We can now, without introducing any constructive difficulties, make 
the radius of the slot (Fig. 127) of any required magnitude, the slot and 

the slider becoming flatter than 
before. If we make the radius 
infinite, and at the same time 
make the link D infinite, that is, 
make the distances cd and ad 
(Fig. 105) both infinite simul- 
taneously, we shall obtain the mechanism shown in Fig. 128, which 
will be called the sliding-block linkage. We thus see that the line of 
connection in a sliding pair is a normal to the sliding surface from the 
axis of the pivot in the slider; and the same is true if a roller is substituted 
for the slide, these appliances replacing the equivalent link so far as the 
motion is concerned. 

In Fig. 128 either of the four pieces A, B, C, or D may be fixed, 
thus giving rise to four mechanisms which will now be considered. 

ii2. The Sliding-block Linkage.— Considering the link D (Fig. 128) 
as fixed, we obtain the mechanism commonly used in pumps and direct- 
acting steam engines. When employed in a steam engine, the block C, 
called the cross-head, is the driver and the crank A the follower; in a 
pump the reverse is the case. 

Movement of Cross-head.— In Fig. 129 let ab represent the crank, be 
the connecting-rod, and mn the path of the point c in the cross-head. 
The travel of the cross-head 
mn is equal to twice the length 
of the crank ab, and the dis- 
tance of c from a varies be- 
tween B-\-A=an and B—A = 
am, A being the length of the 
crank, and B the length of the 
connecting-rod. 

To find the distance the 
point c has moved from n, the ' 

beginning of its stroke or- travel, FlG - 129 ' 

let the angle made by the crank with the line an be represented by 6, 




ss 



LINKWORK. 



and draw bg perpendicular to an. The movement of the cross-head from 
the beginning of its stroke is, for the angular motion 6 of the crank, 

en = an — ac = an — K ag-\- gc) . 



From the right triangle beg, 

gc = 



Hence 



cn=an—ab cos/9 

=A+B-A cosd-VB 



Vbc-bq\ 



v be 2 -ab 2 sin 2 6 



=A(l-cos#) + 5il 



A 2 sin 2 0. . . ... . (27) 

^l-^sin 2 ^}. . . . (28) 



If the length of the connecting-rod has a Certain relation to that of 
the crank, A being the length of the crank, and I A that of the rod, we 
have, substituting IA for B in equation (27) 



cn=A(l+Z-cos#-V7 2 -sin 2 #) (29) 

The motion may be represented graphically by plotting a curve,. 

where the ordinates represent suc- 
cessive values of en, and the abscissas 
represent angular positions of the 
crank ab. Fig. 130 shows the curve 
for the linkage given in Fig. 132. 

L.V. Ratio. — In Fig. 129 continue 
the line of the connecting-rod to s, 
and draw the line as through a and 
perpendicular to am. The instan- 
taneous centre of the rod be is at o, 
found by drawing the lines bo and 
co perpendicular to the lines of 

motion for the instant of the points b and c respectively. 

As the l.v's of the points b and c are proportional to their distances 

from the instantaneous centre o (§ 23), we have 

l.v. of c _oc _as 
= o6 = 




90 u 120' 

Fig. 130. 



l.v. of b 



ab 



as 
~A' 



(30) 



as the triangles abs and obc are similar. 

From the similar triangles cas and cbg we have 



ac 



or as 



, ac 7 aq+qc 

bg—=bg y y : 

gc y gc 



as = 



as_ 
bg~gc 
A sin d\A cos d+ VB 2 -A 2 sin 2 d\ 



\/B 2 ~A 2 sin 2 



THE SLIDING-BLOCK LINKAGE. 



89 



Substituting in (30), we have 



l.v. of c as . n , A sin 6 cos 
l.v. of b A VB 2 -A 2 sin 2 6 



(31) 



The velocity of b being constant, that of c can be found by equa- 
tion (31). 

This same result may be obtained by another method. Since velocity 

ds 
if variable may be expressed by the equation v—-^, we may find the 

l.v. of c by differentiating equation (27), where cn = s and where the 
angle must be expressed in terms of the a.v., a, of the crank A, and 
of the time /. Writing equation (27) in this form gives 



s=A+B-A cos at-\ / B 2 -A 2 sin 2 at. 



' ds . . aA 2 sin at cos at 

.'. l.v. c = ^r- = aA sin at-\- . 

dt VB 2 -A 2 sin 2 at 



But l.v. b = aA; 



l.v. c . , A sin a? cos <x£ 

-, r = sm at+ — ~ — - _ . 

l.v. b VB 2 - A 2 sin 2 at 



(32) 



(33) 



When # = 90°, as = A, and the velocities of c and b are equal. To 
find other values for 6, when the velocities c and 6 are equal, we have, 
from equation (31), 

l.v. of c i . n , A sin cos 6 
= 1 = sm +- 



l.v. of 6 V£ 2 -A 2 sin 2 0' 

(1 -sin 0)\/5 2 -A 2 sin 2 fl=A sin 0\A-sim 



V£ 2 -A 2 sin 2 = 



J. sin 0\/l — sin 5 
l-sin0 



Squaring, 

Solving, for sin 0, we have 



g^wfl= A2si " 2g(1 + sing) . 

1 — sin 



sm 



4A 2 



£±V8A 2 +£ 2 |. 



1.0 — 



The l.v. ratio between b and c may be shown graphically, using 
coordinate axes, the ordinates rep- y 
resenting the ratio and the ab- 
scissas representing angular posi- 
tions of the crank ab. Fig. 131 oi- 

0.3- 

shows the curve for the linkage il'/'' 

given in Fig. 132. • °o° 3 o° eo° 75 90 120 150 iso° 

Fig. 132 illustrates other Fig. 131. 

methods of showing the l.v. ratio. In this figure the constant l.v. of b 




90 



LINKWORK. 



is represented by the crank length ab = A. From equation (30) we 
have 

. l.v. c as 
l.v. b ab' 

Therefore, if we lay off on the line ab, which shows the crank posi- 
tion, the distance at = as, and repeat this construction for a sufficient 
number of crank positions, we shall obtain the full curve ata, where the 
intercept at on the crank line shows the velocity of c, ab being the con- 
stant velocity of b. A similar curve would be found for the crank posi- 
tions below the line ma. Similarly we might obtain the full curve ntjn 
by laying off on the successive perpendiculars drawn through the point c 




Fig. 132, 

the corresponding distances as; then the ordinates of this curve 
drawn through the cross-head position would give the velocity of c 
at that position, the ordinate no = ab representing the constant l.v. 
of b. 

If the length of the* connecting-rod B (Fig. 129) is made infinite, 
the motion of the point c will be equal to the projection of the motion 
of b on the diameter b ± b 2 , and will be, therefore, simple harmonic 
motion. For any angle 6 this motion will be 



b ± g = ab(l — cos 6), 



(34) 



Referring to equation (28) , the motion of c when the connecting-rod is finite 

varies from harmonic motion by the quantity i?( 1 — a/1 — -^ sin 2 #), 

which approaches zero as a limit as B is made greater. 

The connecting-rod is rarely made more than six or seven times 
as long as the crank, as the additional space required is not com- 
pensated for by the slightly nearer approach to harmonic motion. 
The dotted curve in Fig. 130 would be the curve for harmonic 
motion. 



THE SLIDING-BLOCK LINKAGE. 



91 



The l.v. of c, if B were infinite, would be found by resolving the l.v. 
of b into two components, one 
vertical and one horizontal 
(Fig. 133), the horizontal com- 
ponent being the l.v. along the 
rod be if it were infinite and 
thus equal to the l.v. of c. 
From this we have 

l.v. c 




l.v. b 



bb" 
W 



sin 6. 



(35) 



Comparing this with equation (31), the l.v. ratio when B is finite varies 

from harmonic by the amount _ , which approaches zero as 

VB 2 — A 2 sin 2 6 

a limit as B is made greater. The dotted curves in Figs. 131 and 132 

would show the l.v. of c if the motion were harmonic. In Fig. 132 the 

dotted curves will be found to be circular. 

If, in Fig. 128, we consider the crank A as the driver, it can always 

produce reciprocating motion in the block C; but if C is the driver, it 

cannot produce continuous circular motion unless some means of pass- 





A 


Bi 












i — L! — i 




i 






1 






C 


C] 


A 






1 


1 




1 — i — ' 







Fig. 134. 



Fig. 135. 



ing the dead-points be devised. This is usually accomplished in 
steam engines by attaching to the crank-shaft a heavy fly-wheel, the 
momentum of which carries the crank by the dead-points. The impossi- 
bility of starting at the dead-points still remains. 

To obviate this difficulty two crank and connecting-rod mechan- 
isms may be combined, as shown in Fig. 134, where the cranks are 
placed at right angles to each other and joined by a shaft. This combi- 
nation is employed in locomotives, and in hoisting and marine engines, 
one crank being very near its best position to be acted on by the rod 
while the other is at a dead-point. 



92 



LINKWORK. 




Fig. 135 shows another method of passing the dead-points sometimes 
used in marine engines. Here the two connecting-rods B and B ± are 
located in parallel planes and act upon the same crank A. By suitably 
forming the ends of the rods, they might be located in the same plane. 
113. The Swinging-block Linkage. — If we consider the link B 
^ ^ h (Fig. 128) as fixed, we ob- 

tain the swinging-block 
linkage commonly used 
in oscillating engines. 
Such a mechanism is 
shown in Fig. 136, from 
which we obtain the 
mechanism shown in Fig. 
137 by inverting the pair 
CD, this having no effect 
on the relative motions 
ofC andD(§ 39). Here 
A represents the crank, 
D the piston-rod, and C, 
the swinging slide or block, the cylinder. 

In Fig. 138, which represents an oscillating engine, three relations 
may be studied: 1° the motion of the piston D relative to the cylinder 



Fig. 136. 




Fig. 137. 




Fig. 138. 

C; 2° the ratio of the l.v. of the piston relative to the cylinder and 
the l.v. of b; and 3° the a. v. ratio of the cylinder, about the axis c of 
the trunnions which support it, to the crank A. 

1° To find the distance dn (Fig. 138) which the piston has moved 
from the beginning of its stroke, for a given angle bac = d. Let e be 
the point on the piston-rod D which is coincident with c when A and D 
are in the same line; then ce will be equal to the motion of the piston dn.. 
From the figure we have 



But 



dn = ec=bc—be=bc — (B—A). 

6c=V A 2 + B 2 -2AB cos 0; 
dn = ec=VA 2 +B 2 -2AB ca&D-B+A. . 



(36) 



THE SWINGING-BLOCK LINKAGE. 



93 



2° To find the l.v. of the piston in the cylinder. In Fig. 139 let 
W represent the l.v. of b around a; then the component bb" along the 
piston-rod would be the 
desired l.v. of the piston 
relative to the cylinder. 
The ratio of the l.v. of 
piston in the cylinder to 
the l.v. of the crank-pin 
b may be found by 
reference to the instan- 
taneous axis of the link 
bd. In the link bd the 
direction of the motion 
of b will be along the line 
W, and the direction of 
the point on the link 
which coincides at the 
instant with the axis c FlG - 139 - 

of the trunnions which support the cylinder will be along the line bd; 
therefore the instantaneous axis of bd will be found at o, the intersection 
of the lines bo and co perpendicular respectively to W and be. 




l.v. of piston relative to cylinder _co 
l.v. of crank-pin b bo 



bb^ 
bb" 



since the triangle obc is similar to Wb" . 

To find the actual l.v. of any point in the piston-rod or piston, as d, 
we have 

l.v. of d _do 

l.v. of b~b~o' 

and its direction of motion would be along dd' perpendicular to do. 

3° To find the ratio of the a.v. of the cylinder about the axis c, and 
the a.v. of the crank ab. To determine this ratio, by using the law 
deduced in § 98 for a.v. ratio, it will be necessary to draw the centre 
lines of the two infinite links which have been replaced by the sliding 
pair. These lines must be perpendicular to the centre line of the sliding 
pair (§ 111) and will be be and co (Fig. 139). Since ac is the line of 
centres, be will be the centre line of the infinite connecting link. Apply- 
ing the law for a.v. ratio, 

a.v. cylinder ae af 
a.v. ab be cf 

114. Quick- return Motion using the Swinging-block Linkage.— 

If in Fig. 136 or 137 the piece C is made long, and the link D is reduced 
to a sliding block, we have a linkage which may be drawn as shown 



94 



LINKWORK. 



in Fig. 140, where a uniform rotation of A will cause an oscillation of 
the link C. If A turns R.H. through the angle a, the link C and the 
connected tool-slide E will travel to the right; and while A turns through 
the angle /?, C and E move to the left; thus we have 

time of advance of E a 

T 



time of return of E 



If we have given the desired time-ratio of advance to return; the 
line of centres, ac; the centre, a; and the length of the crank, ab; to 

locate the centre 
c draw the crank- 
pin circle bb t and 
make the angle b t ac 
equal to |/3, where 
a _ advance 




The 



Fig. 140. 



return 
tangent b x c to the 
crank-pin circle at 
b x will give the de- 
sired centre c. 

To determine the 
l.v. of the slide E at 
any moment, given 
the l.v. of b around a, 
let bb' represent the 
l.v. of b around a, 
then bb" will be the 
l.v. b around c; dd f 
will be the l.v. of d 



around c, and ee' will represent the l.v. of the slide E. 



To find the ratio 



a.v. of C 



we have, as in Fig. 139, the sliding pair 



a.v. of A 

replacing two infinite links, the centre lines of which must be perpen- 
dicular to the centre line of the sliding pair. Therefore the centre 
line of the infinite connecting link would be the line bk through the 
crank-pin b perpendicular to the centre line cd of the sliding pair. This 
would give 

a.v. C _af _ag 

a.v. A be eg' 

When the crank is in the positions ab 1 and ab 2 it will be seen that the 
centre line of the infinite connecting link would pass through a, giving 

, = — . or the link C has no angular motion, as would have been 
a.v. A b^ to 

evident from its position. From the position 6 X the a.v. of C increases 



THE TURNING-BLOCK LINKAGE. 



95 



to a maximum, for the forward stroke, at the position b 3 ; the maximum 
on the return stroke being at b 4 . In the former position 

a.v. C _ab 3 

a.v. A cb 3 



and in the latter position 



a.v. C _ab i 
c6 4 * 
-If the link A, Fig. 



a.v. A 
115. The Turning-block Linkage 

sidered as fixed, we shall have the linkage shown 
in Fig. 141, where B is a crank turning uniformly 
about 6, which on so turning will cause D to 
make complete rotations about a, but with a 
variable motion. 

Whitworth Quick Return is the name given 
to the linkage when it is used as a quick-return 
motion, as in Fig. 142. If the crank be (Fig. 
142) turns uniformly R.H. from the position c x to 
the position c 2 , the slide e will travel from its 
extreme position at the right to the end of its 
stroke at the left; and while be turns from c 2 to c x 

time of advance of e a 

J 




128, is con- 



\ 



Fig. 141. 

the slide e returns: 



time of return of e 

To locate the centre a, given the time-ratio of advance to return; 

the line of centres, the axis jb 

and the crank be; make the 

angle cfia equal to J/?, where 

a advance . 

and draw cm 




FlG; 142. 



p return 

through c x perpendicular to the 
line of centres; the point a 
is the axis of the link ad. If 
the stroke of the slide e is not 
on a line passing through a, 
but below it, as is commonly 
the case, the time-ratio of advance 
to return would be somewhat different from the above. 

The l.v. of the slide e may be found by the same method as that 
used in the swinging-block linkage. 
For the a.v. ratio we have 

a.v. ad _bf bg 
a.v. be ac~ ag' 
the line eg being the centre line of the infinite connecting link. This 
ratio is unity at the two positions q and c 2 , the centre line of the con- 
necting link being parallel to the line of centres. From the position c t 



96 



LINKWORK. 



bc 3 




the ratio diminishes until c is at c q , when it becomes — - 3 , which is its 

ac 3 

minimum value; it then increases to unity at c 2 , still further increasing 

be 
until at c 4 it has its maximum value — 4 . 

ac 4 

For the development of this linkage as practically used see § 118, 

Fig. 151. 

116. By fixing the block C (Fig. 128) we obtain the fourth form 
of the linkage having one sliding pair, as shown in Fig. 143. The link 

fc.v^ B now swings about a fixed axis in C, 

and the slide D moves rectilinearly to 
and fro in the block C, which is now the 
frame; the link A, now a connecting- 
rod, has a complex motion made up 
of a combined oscillation and rotation. 
If the link A is so expanded that it can 
be caused to make complete rotations relative to the axis a, the link D 
would have a reciprocation relative to the block C, the stroke of D being 
twice ab. Such a development is shown in Fig. 152, § 118. 

117. The Isosceles Sliding-block Linkage. — If we make the length 
of the connecting-rod B (Fig. 128) equal to that of the crank A, we 
shall obtain the linkage abc (Fig. 144). 
Here, if we consider ab as the driver, 
and c to start from the position c v it 
will be found that when the crank ab 
is at an angle of 90° with ac t , the path 
of c, the block c is directly over a, and 
any further rotation of ab will only cause 
a similar rotation of be. In order to 
cause c to continue in its path when 
ab reaches the 90° position, it will be 
necessary to find the centroids of B 
and D, and apply the principles of § 105. 

If we assume the piece D as fixed, 
the centroid of B will be the light circle e x e^c 2 e 2 , the point b moving 
in the circular path bfib 2 , and c in the rectilinear path c 1 cc 2 . The trace 
of the surface for be which by rolling on the centroid of be would give 
the same motion to be as the linkage, would be found, by the method 
of § 27, to be the smaller circle coea. Now if we continue the line cb 
to e, making be = be, the point e will be found at e 1 and e 2 when ab and 
be are in line; and if we supply the centroids with pairs of elements 
which will be in contact at e 1 and e 2 , these pairs will cause the point c 
to -travel by a. The motion of c will then be four times the length of 
the crank ab. 




Fig. 144. 



THE ISOSCELES SLIDING-BLOCK LINKAGE. 97 

Since the point c in the sliding block is always under the point o, 
and since o is always in the continuation of ab, if ab turns uniformly 
the block C will have simple harmonic motion. This can also be shown 
as follows: 

l.V. C CO CO _ ■ . A 

i — *: = *r = i — =2 sm v- 
l.v. 6 6o |oo 

.*. l.v. c=2Xl.v. OX sin (9, 

or the l.v. c is of the same nature as that illustrated by Fig. 133, but 

twice as fast, assuming the same length of crank and the same velocity 

of crank-pin. 

The four mechanisms of the linkage with one sliding pair here become 
two only, the first of which has just been discussed. The same mechanism 
is obtained whether we consider the block C or the slot in D to be sta- 
tionary. The second case is when the link ab is fixed; or the same 
mechanism would result by fixing the link be. 

If we fix the link ab, the link be, rotating about b, would cause the 
link ac v to turn about a, and if the centroids are used the small circle 
coe, containing be, turning uniformly around 6, would cause the large 
circle c 1 oe 1 , containing ac x , to make complete rotations about a, the 
a.v. ratio being 

a.v. D _bo _1 
a.v. B ao 2 ' 
the radius of the smaller circle being one half that of the larger. 

The a.v. ratio can also be shown from the linkage. In the given 
position the centre line of the infinite connecting link will be co; there- 
fore 

a.v. ac 1 _bf _oo_l 
a.v. be ac ao 2 ' 

It is interesting to note that the path of the point e relative to ac t 
is a diameter of the circle c x e x c 2 e 2 at right angles to c x c 2 , and if we replace 
D by a disc having two grooves at right angles to each other, inter- 
secting at a, at the same time supplying e with a sliding block similar 
to c, the disc D will make one revolution while ce makes two revolu- 
tions. Thus ce can be considered as a wheel of two teeth rolling inside 
of another, D, of four teeth: in such a case, the blocks c are usually made 
cylindrical, and roll in the grooves so as to reduce the friction. Three 
grooves might be made in the disc intersecting at a, and making angles 
of 60° with each other; the circle ce would then need to be supplied with 
three rollers spaced equidistant on its circumference ; the relative motion 
of the disc and be would be the same as before. 

1 1 8. Expansion of Elements in the Linkages with One Sliding Pair. — 
So far we have not concerned ourselves with the diameter of the cylin- 
dric pairs in these mechanisms, as alterations in the diameters would 
not affect the relative motions. Also a change in shape or size of the 



98 



LINKWORK. 



links will not alter the relative motions, so long as the centre lines of 
the elementary links remain unchanged, and yet such change may 
make the action of the linkage possible. Since these enlargements of 
the elements of the cylindric pairs sometimes conceal the real nature 
of the mechanism and cause much indistinctness, it will be well to con- 
sider a few cases here. 

We will consider first the sliding-block linkage, shown in Fig. 128. 




Fig. 145. 

Each of its four links is more or less closely connected with its three 
cylindric pairs, and their forms are therefore dependent upon the relative 
sizes of the latter, although this size does not affect the nature of their 
relative motion. Evidently we do not alter the combination kinemat- 
ically, if we increase the diameter of the crank-shaft so as to include 
the crank-pin as shown in Fig. 145, where the different links are 
lettered the same as in Fig. 128. The open cylinder of D must be 
enlarged to the same extent as the shaft, so that the pair is still 
closed. 

This arrangement is used in practice, in some slotting and shearing 
machines ; to work a short-stroke pump from the end of an engine shaft ; 
and in other cases where a short crank forms one piece with its own 
shaft. 

If we expand the crank-pin until it includes the shaft, as shown in 
Fig. 146, we obtain the common eccentric and rod, which can be seen to 
differ only in form from the common crank and connecting-rod. This 

mechanism is much 
used to operate the 
valve motions in 
steam engines , 
where it is neces- 
sary to obtain a 
_ reciprocating mo- 
tion, often less than 
the diameter of the 
FlG U6 engine shaft. The 

part of the rod B 
which encloses A is called the eccentric-strap, and is made in two parts, 




EXPANSION OF ELEMENTS IN LINKAGES. 99 




A-H 



i-r 



LJLI -A 



Fig. 147. 




,— , C 



m 



sr 



Fig. 148. 



LofC. 



100 LINKWORK. 

and separate from B, the eccentric-rod, which is usually bolted to one of 
these parts; the cylindrical pair is also so shaped as to allow no axial 
motion of B on A. 

If the crank-pin is still further expanded until it includes the cross- 
head pin, we shall obtain the arrangement shown in Fig. 147. In this 
case, the element of the cylindric pair which belongs to the crank A has 
been inverted, and thus made open (§ 39). The rod B becomes an 
eccentric disc which swings about the bearing in the 'piece C, and is 
always in contact with the hollow disc A, carried by the shaft turning 
in D. 

If, instead of enlarging the crank-pin to include the cross-head pin, 
we enlarge the latter to include the former, the arrangement shown in 
Fig. 148 is obtained. The rod B is again an eccentric disc or annular 
ring; but it now oscillates in a ring forming part of the piece C, 
while the crank-pin drives it by internal contact. In order to make 
the relations of these expansions more clear, fine light lines have been 
drawn in each case, showing the elementary links. The above exhausts 
all the practicable combinations of the three cylindric pairs. 

In Fig. 148 we can replace the link B by an annular ring containing 
the crank-pin and oscillating in a corresponding annular groove in the 
piece C. So long as we keep the centre of this ring the same as that of 
B } we have not altered the mechanism, and as the motion of the ring is 




Fig. 149. 

merely oscillatory, we need only use a sector of it, and enough of the 
annular groove to admit of sufficient motion of the sector in its swing. Fig. 
149 represents the arrangement altered in this way, the different parts 
being lettered the same as in Figs. 148 and 128; B is still the connecting- 
rod, and its motion as a link in the chain remains the same as before, and 
is completely restrained; the shape of the sector always fixes the length 
of the connecting-rod. This mechanism is made use of in the Stevenson 
and Gooch reversing-gears for locomotives, and in other places; the 
chains are not there simple, but compound. 

The mechanism shown in Fig. 150, which sometimes occurs in slotting 



EXPANSION 'OF ELEMENTS IN LINKAGES. 



101 




Fig. 150. 



and metal-punching machines, is another illustration of pin expansion. 

The whole forms a sliding-block linkage; the 

link B is formed essentially as in Fig. 149, but 

here the profiles against which it works are con- 
cave on both sides of the crank-pin, the upper 

profile being of large, and the lower of very small, 

radius, but both forming part of the block C. 

The work is done when the block C is moving 

downwards, and the small radius profile being 

then in use, the friction is reduced. In this case 

the block C is so enclosed by the slide D that 

the profiles representing the cross-head pin lie 

entirely within the sliding pair, an illustration of 

how the method of expansion can be applied to 

the fourth or sliding pair. 

To make possible the action of the Whitworth Quick Return, § 115, 

Fig. 142, the axis b may be expanded until the axis a can be placed upon 

it, or a shaft passed through it if 
the crank ad is required on the 
other side of the frame. Fig. 151 
shows the expansion where A is the 
enlarged axis b and is part of the 
frame of the machine; the crank be 
has become a spur-gear B, turning 
on A, and driven uniformly by a 
small pinion below it. The crank- 
pin works in a block C turning on 
the face of B and fitting in a slot 
in the crank D, the expansion of the 

link ad, pivoted on a pin a provided for it in the frame A. While the 

gear B turns so that the axis of the crank-pin moves through the 

larger angle cfic 2 , the slide E will have its slow motion or advance, the 

return occurring while the gear moves through the smaller angle cjbc v 
A development of the fourth form of the linkage with one slide, 

mentioned in § 116, Fig. 143, is shown in Fig. 152. The connecting 

link ab is expanded into a 

worm-wheel A, which may be 

made to rotate about the axis 

a by a worm keyed to the 

shaft D. The worm and wheel 

are kept in contact by a piece 

which supports the bearing of 

A, hangs from the shaft D, 

and confines the worm between 

its bearings. A rotation of the shaft D will turn A, causing a reciproca- 




Fig. 151, 




Fig. 152. 



102 



LINKWORK. 




Fig. 153. 



tion of the axis a, and consequently of the driving shaft D, through a. 

distance a 1 a 2 equal to twice ab. 

A change in the shape of an elementary link frequently per-^ 

mits motions to take 
place which are not 
otherwise possible. In 
Fig. 153, for example, 
a complete rotation of 
A to cause a reciproca- 
tion of C would be pos- 
sible with the open rod 

B moving around the fixed shaft E, but not with elementary link be, 

shown by a light line. 

1 19. Linkwork with Two Sliding Pairs. — If we apply the principle- 

of § 111 to Fig. 149, and allow the length of the link B to become 

infinite, the slot in the 

slide C will become 

straight and at right 

angles to the sliding 

pair C, and the con- 
necting block B becomes 

a prismatic slide. 

Such a mechanism 

is shown in Fig. 154, 

the different parts being 

lettered the same as in 

Fig. 149. The block C 




Fig. 154. 



now consists of two sliding pairs at right angles to each other, and the 

connecting-rod is infinite in length. Here, as in the linkage with one 

slide, since there are four ele- 
mentary links, four mechan- 
isms would result. It will be 
seen, however, that two are 
identical, leaving three distinct 
mechanisms. Fig. 155 cor- 
responds with Fig. 154 with- 
out the expansion of the cross 
C, which would be required to 
make the mot on possible, and 
will be more convenient to use 
in studying the relative motion. 
120. This mechanism with 
the link D fixed is often used, 

and is known under various names, as crank and slotted cross-head, crank 

with an infinite connecting-rod, harmonic motion, etc. 




Fig. 155. 



LINKWORK WITH TWO SLIDING PAIRS. 



103 



sin 6 



(38) 



If in Fig. 155 we assume the block D fixed, and give a uniform rota- 
tion to A , we shall have a reciprocation of the cross C through D while 
the block B slides up and down on the cross, giving to it a simple har- 
monic motion. If the bloc B were fixed instead of D, the cross C would 
Iiave exactly the same form of motion as when D was fixed, only it 
would be up and down through the slide B. 

To determine the motion of C for any angular motion of A, let the 
crank A turn through the angle 0, Fig. 155; the distance cd through 
which the cross has moved will be 

cd=ac — ad=ab{\ — cos 6), (37) 

which was found in § 112, equation (34), as the equation for simple 
harmonic motion. 

For the l.v. ratio, from Fig. 155, 

l.v. of cross C _bf 
l.v of crank-pin b be 
(see equation (35), § 112). 

Fig. 156 shows a combination of an eccentric circular disc A, and 
a sliding piece C, moving through fixed guides, 
one of which is shown at Z). A uniform rota- 
tion of A about the axis a will give harmonic 
motion to C. This can be shown by noticing 
that the distance which C has moved from its 
highest position is 

cd = ef = ab(l — cos d), 

which is the equation for simple harmonic 
motion where 2ab is the stroke of the slide. 
This mechanism can also be .found by an 
expansion of the crank-pin b, Fig. 154, until 
it includes the shaft a, the slot in C being cor- 
respondingly enlarged, and then after turning the figure through 90°, 
omitting the lower part of the cross C, allowing A and 
C to be paired by force- closure. 

The Swash-plate. — The apparatus shown in Fig. 
157, known as a swash-plate, consists of an elliptical 
plate A set obliquely upon the shaft S, which by 
its rotation causes a sliding bar C to move up and 
down, in a line parallel to the axis of the shaft, 
in the guides D, the friction between the end of the 
bar and the plate being lessened by a small roller 0. 
When a roller is used, the motion of the bar C is approxi- 
Fig. 157. mately harmonic— the smaller the roller the closer the 
approximation. If a point is used in place of the 
roller, the motion is harmonic, which can be shown as follows: 




Fig. 156. 




104 



LINKWORK. 



Since the bar C remains always parallel to the axis of the shaft, the 
path of the point 0, projected upon an imaginary plane through the 
lowest position of and perpendicular to the shaft S, will be a circle, 
and the actual path of on the plate A will be an ellipse. 

In Fig. 158 let eba represent the angular inclination of the plate to 
the axis of the shaft, ab the axis of the shaft, eof the actual path of the 
point o on the plate, and the dotted circle erd the 
projection of this path upon a plane through e 
(the lowest position of o) perpendicular to the 
axis ab. 

Draw om perpendicular to ef, or perpendicular 
to the plane erd, and rn perpendicular to ed, the 
diameter of the circle erd. Join ran, and suppose 
the plate to rotate through an angle ear = 6, and 
thus to carry the point o through a vertical distance equal to or. 

Then 




Fig. 158. 



or = mn = 



en ( mn en\ 

abX— ( as — j- = — I 

ea \ ab eaj 

H?) 



= ab 

= a6(l — cos d), 
or the same formula as was derived in the case of harmonic motion. In 
this case ab represents the length of the equivalent crank, and is equal 
in length to one-half of the stroke of the rod C. 

121. If instead of fixing the block D or the block B, Fig. 155, we as- 
sume the link A to be fixed, we obtain the second form of the mechanism. 

This is shown in Fig. 159, 
where the axes a and b are 
fixed and the blocks B and D 
are free to turn, while the 
cross C will slide through them, 
being forced at the same time 
to revolve. If D turns uni- 
formly, B will also turn uni- 
formly; for if D turns through 
any angle 6 as shown, the cen- 
tre lines of the slots in the 
cross C must occupy the posi- 
tion shown by the light lines, 

and the angle abc has dimin- 
Fig. 159. . . fe 

ished by the same amount as 

bac has increased or B has turned through the same angle as D. It is 




LINKWORK WITH TWO SLIDING PAIRS. 



105 



interesting to notice that the piece C has an eccentric revolution , the 
intersection c of the centre lines moving in a circular path with ab as its 
diameter. 

Oldham's Coupling. — The mechanism shown in Fig. 160, known as 
Oldham's coupling, is an interesting example of the above mechanism. 
Its object is to connect two parallel shafts placed a short distance apart 
so as to communicate a uniform rotation from one to the other. 




Fig. 160. 



The bearings for the two shafts a and b are in the piece A, which 
takes the place of the crank A (Fig. 159) ; the pieces B, C, and D (Fig. 160) 
take the places of those similarly lettered in Fig. 159, and are drawn 
separated at the right of the figure to make their construction clearer. 
The piece C has two diametrical slides c and d placed on its opposite sides 
and at right angles to each other. The grooves c t and d v in the pieces 
B and D respectively, fit the corresponding slides similarly lettered on C. 
Fig. 160 is an expansion of the elementary links in Fig. 159. 

122. If in the linkage with two slides, Fig. 155, we fix the cross C, we 




Fig. 161. 

have the third mechanism. This is shown in Fig. 161, where, if we 
replace the cross by grooves, the blocks B and D may slide in their re- 
spective grooves, with the result that any point on ef, between e and / 



103 LINKWORK. 

or on ef produced, will trace an ellipse. The point m will trace an ellipse 
of which the semi-major axis is fm and the semi-minor axis is em. The 
point n will trace an ellipse, of which the semi-major and semi-minor 
axes are fn and en respectively. The linkage thus arranged is called an 
elliptic trammel. All ellipses traced by points on ef beyond e have the 
difference of their semi-major and semi-minor axes equal to ef. All 
ellipses traced by points between e and / have the sum of the semi- 
major and semi-minor axes equal to ef. The point m t half-way between 
e and / traces a circle with a diameter ef (see Fig. 144). In the elliptic 
trammel the ellipse is usually traced by a point outside of e; e and / 
are made so that their distance apart is adjustable and they are set one 
half the difference of the major and minor axes apart. 

An ellipse can be readily drawn by taking a card one corner of 
which shall represent the tracing-point n. Points corresponding to 
the desired positions of e and / (Fig. 161) are then marked on the edge 
of the card, and by placing these points in successive positions on lines 
at right angles with each other, corresponding to the slots in which the 
blocks in Fig. 161 move, and marking the successive positions of n, will 
give a series of points on the required ellipse. 

To prove that the point n moves on an ellipse, let np = x and nr = y; 

nf (semi-major axis) =a and ne (semi-minor axis) = fr. The equation for 

x^ y^ 
an ellipse referred to the centre as the origin is -2+p" = l« 

In Fig. 161 we have 

x np . y nr 

— = -f and -?=— , 
a nf b ne 

and, since the triangles em and npf are similar, 

o o o o — — O 

np nr _np fp _nf 

— j.2 ' TT~2 ~~ —£2. ' l =^2 _ == $ = 1 } 

nf ne n f n f n f 
or 

x i + i= l < < 39 > 

and the point n moves on an ellipse. 

Now in drawing an ellipse, the paper is fixed, and the pencil is moved 
over it; but in turning an ellipse in a lathe, the tool, which has the same 
position as the pencil, is fixed, and the piece to be turned should have 
such a motion as would compel the tool to cut ellipses. 

This is accomplished in the elliptic chuck, Fig. 162, the points / and 
e, which correspond with the points / and e (Fig. 161), being fixed, and 
the tool acting at the point n, Fig. 162, corresponding with the tracing- 
point n in Fig. 161. The sliding blocks BB and DD, as well as the 
points /, e, and n, correspond with the parts similarly lettered in Fig. 
161, but in this case ef is fixed instead of C. 



LINKWORK WITH TWO SLIDING PAIRS. 



107 



In the form of elliptic chuck shown in Fig. 162, A represents the 
headstock casting, which furnishes bearings for the spindle E, the trace 
of the axis of which is shown at e, and a fastening for the piece GG, 
which provides a centre / through which the centre line of the slot B'B' 
always passes. The face-plate C, to which the material to be turned 



m 
t — ' 


H 

■it 

A 


j 
!• 

IJ-- 




Fig. 162. 



is attached, is furnished with two straight grooves at right angles to 
each other. The lathe-spindle E is provided at its front end with a 
straight arm DD, the centre line of which passes through its axis e. 
This arm fits the groove D'D' and communicates the rotation of the 
spindle to the face-plate C, at the same time always compelling the 
centre line of the slot to pass through e. 

The piece GG, consisting of an annular ring with two projecting lugs, 
is arranged to slide in the line fe, and is held in place by bolts fastening 
it to slotted projections on the headstock casting FF. This annular 
ring allows the spindle to turn within it and carries the strap HH, which 
can turn upon it but can have no axial motion, as will be seen from its 
construction. Two projections BB, the centre line of which passes 
through /, the centre of the ring, are formed on the strap HH, and are 
fitted to the dovetailed groove B'B' in the piece C; thus the connec- 
tion between the pieces G and C is such that the centre line of the groove 
B'B' always passes through the point /, and no motion along the axis 
of the spindle is allowed in C. 

The above is then an arrangement where the plate C has a motion 
such that the centre lines of the two grooves always pass through the 
points e and /. The distance ef, equal to the difference between the 
semi-axes of the ellipse, can be adjusted at will, and various ellipses may 



108 



LINKWORK. 




be turned. For example, if the axes of the ellipse are 6" and 4", the 
distance ef is 1". 

123. In the linkages with one slide, the centre line of the sliding pair 
may not pass through the point a, as in Fig. 128. This will give rise 

to another series of 
mechanisms, some- 
what similar to those 
described. 

In the sliding- 

block linkage, Fig. 

163, the motion of the 

block C from one end 

of its stroke to the 

other will require a 

motion of less than 

180° of the crank ab 

in one direction, and 

FlG - 163 ' more than 180° in the 

other. Slight differences would also occur in the l.v. ratio, but the same 

laws apply, the instantaneous centre being at and 

l.v. c co _as 
l.v. b bo ab' 
In the swinging-block linkage the mechanism could be arranged as 
in Fig. 164. In the position shown in the figure 

a.v. cd _ae 
a.v. ab cf ' 

since be is the centre line of the infinite connecting 
link. To prove that this is so, let bg represent the 
l.v. of b around a. Resolve this into two com- 
ponents, one of sliding along cd and the other of 
rotation of b around c. This will give bh as the 
l.v. of b around c. We may then write 

l.v. b around c bh 
a.v. cd be be 

w 

ab 



be 
a.v. ab l.v. b around a 



ab 

but by the similar triangles bhk and cbf, and bgk 
and abe, 




Fig. 164. 



bh bk . bq 

t— = -t and -Z- ■- 
be cf ab 


_bk m 
ae ' 


a.v. cd bk ae 
a.v. ab cf bk 


ae 



THE CONIC FOUR-BAR LINKAGE. 



109 



Similarly in the linkages with two slides the arms of the cross C 
may not be at right angles, in which case a new series of mechanisms 
will result, one example being shown by Fig. 165. 

In Fig. 165 let be represent the l.v. of b around a, then bf will be the 
l.v. of the cross C. If the crank coincides with the centre line of the 
slot in C, the l.v. of C, bj x at ab lf will be greater than the l.v. of the crank- 
pin, b x e v If ab turns uniformly, the motion of C will be found to be 
harmonic, but the length of stroke is greater than twice ab. 



Z^-T' 





Fig. 165. 

Other forms of the linkage with two slides occur, as in Fig. 166, 
where a slight period of rest is desired for the piece C at the end of each 
downward stroke. To find the l.v. of C at any point, as when the crank 
is at ab, it is necessary to resolve the l.v. of b, represented by be, into 
two components, one, bf, in the direction of motion of the piece C, and 
the other, bg, tangent to the centre line of the slot in C. 

124. The Conic Four-bar Linkage. — If the axes of the four cylindric 
pairs (Fig. 101) of the four-bar linkage are not parallel, but have a common 
point of intersection at a finite dis- / 

tance, the chain remains movable and 
also closed (Fig. 167). The lengths 
of the different links will now be 
measured on the surface of a sphere 
whose centre is at the point of inter- 
section of the axes. The axoids will 
no longer be cylinders, but cones, 
as all the instantaneous axes must . 
pass through the common point of 
intersection of the pin axes. . ' 

The different forms of the cylindric linkage repeat themselves in 
the conical one, but with certain differences in their relations. The 




Fig. 167. 



110 



L1NKWORK 



principal difference is in the relative lengths of the links, which would 
vary if they were measured upon spherical surfaces of different radii, 
the links being necessarily located at different distances from the centre 
of the sphere in order that they may pass each other in their motions. 

The ratio, however, between the length of 
a link and its radius remains constant for 
all values of the radius, and these ratios are 
merely the values of the circular measures 
of the angles subtended by the links. In 
place of the link lengths, we can consider 
the relative magnitudes of these angles, 
which can be also designated by the letters 
A, B, C, and D. 

The alterations in the lengths of the 
links will now be represented by correspond- 
ing angular changes. The infinitely long 
link corresponds to an angle of 90°, as this 
gives motion on a great circle which corre- 
sponds to straight-line motion in the cylin- 
dric linkages. 

Fig. 168 shows plan and elevation 
of a conic four-bar linkage abed, the link 
ab turning about a, and, for a complete 
turn, causing an oscillation of the link cd 
about d through the angle 6, shown in the 
elevation. In the figure each of the links 
be and cd subtends 90°, while the link ab subtends about 30°. Varying 
the angles which the links subtend will, of course, vary the relative 
motions of ab and cd. 

125. Hooke's Joint. — If in Fig. 168 each of the links ab, be, and 
cd is.. made to subtend an angle of 90°, we shall find that ab and cd 
will each make complete rotations. This 
mechanism, known as a Hooke's joint, is 
represented by Fig. 169; a and d are the 
two intersecting shafts, and the links ab 
and cd, fast to the shafts a and d respect- 
ively, subtend 90°, while the connecting 
link be also subtends 90°. 

In order to make the apparatus stronger 
and stiffer, two sets of links are used, and 
the link cb is continued around as shown, 
thus giving an annular ring joining the 
ends of the double links cdc f and bob' . This 
ring is sometimes replaced by a sphere into which the pins c, 




Fig. 168. 




Fig. 169. 



and 



HOOKE'S JOINT. 



i 
111 



b' are fitted, or by a rectangular cross with arms of a circular section 
working in the circular holes at b, c, c' ', and b'. Or, the arms bab 1 and 
cac t may be paired with grooves cut in a sphere in planes passing through 
the centre of the sphere and at right angles to each other. Such forms 
of Hooke's joint are now on the market and much used. 

Relative Motion of the two Connected Shafts. — Given the angular motion 
of ab, to find the angle through which cd turns. Fig. 170 shows a plan 
and elevation of a Hooke's joint, so drawn 
that the axis a is perpendicular to the plane 
of elevation. If the link ab is turned through 
an angle d, it will be projected in the posi- 
tion ab v The path of the point c will be on 
a great circle in a plane perpendicular to the 
axis d, which will appear in the elevation as 
the ellipse bee. The point c will then move 
to c v found by making the angle b 1 ac 1 equal 
to 90°, for the link be subtends 90°, and 
since the radius from b to the centre of the 
sphere is always parallel to the plane of 
elevation, its projection and that of the 
radius from c will always be at right angles. 
The projected position of the linkage after 
turning a through the angle 6 will be ab^d. 
To find the true angle through which the 
link cd and the shaft d have turned, swing 
the ellipse bee with the axis d, until d is 
perpendicular to the plane of elevation, 
when the points c and c x will be found at 
c f and c/, respectively, giving the angle 




Fig. 170. 



c 1 , ac / = <f) as the true angle through which 

the axis d has turned. Or the arm dc 1 may 

be revolved until shaft d is perpendicular to the horizontal plane, giving 

c 1 bc l , = cj), as shown in the plan. 

It is evident from the above that two intersecting shafts connected 
by a single Hooke's joint cannot have uniform motions. If, however, 
two joints are used to connect two parallel or intersecting shafts, they 
may be so arranged that they will have uniform motions. 

Double Hooke's Joint. — Tivo parallel or intersecting shafts may be 
connected by a double Hooke's joint and have uniform motions, provided 
that the intermediate shaft makes equal angles with the connected shafts, 
and that the links on the intermediate shaft are in the same plane. Fig. 
171 gives a plan and elevation of two shafts so connected, and the posi- 
tion after turning through an angle 6. It is evident that one joint just 
neutralizes the effect of the other. 



112 



LINKWORK. 



The term universal joint is often used to designate the above-de- 
scribed mechanism. 




Fig. 171. 
Angular Velocity Ratio in a Single Hooke's Joint. — Fig. 172 reproduces 

the elevation given in Fig. 170, which 
shows the angles 6 and <£> through which 
a and d move respectively. The angle 
eac = a will be the true angle between 
the planes in which the paths of the 
points b and c lie; to find the angle <f> 
analytically in terms of 6 and a, we 
have, from Fig. 172, 




tan <h = -i-Z= C -±t; tan 6 = ^. 
ag ag af 



2l X — 
ag cj 



af 



ac 



= cos a ; 



ac 
ag ac' ae 

.-. tan (f> = tan 6 cos a (40) 

To obtain the velocity ratio, we must differentiate equation (40), 
remembering that cos a is a constant; then 



d<j>_sec 2 6 
dd ~ sec 2 (j> 



cos a 



1 +tan 2 d 
1 +tan 2 <}> 



cos a. . . . . . (41) 



HOOKE'S JOINT. 113 

If we eliminate <j> and 6 from equation (41), by use of equation (40) 

we shall obtain 

dj> = cos a 

dd l-sin 2 0sin 2 a y } 

_ 1 — cos 2 <j> sin 2 a 

— : - (4o) 

cos a 

Assume ab and cd the starting positions of the arms ab and cd respec- 
tively; then equations (42) and (43) will have minimum values when 
sin# = and cos $ = 1; this will happen when 6 and </> are 0° and 180°, 

giving ^- = cosct in both cases. Thus the minimum velocity ratio 

occurs when the driving arm is at ab and ab 2 , the corresponding posi- 
tions of the following arm being cd and c 2 d. Maximum values occur 

when sin# = l and cos <i> = 0; then ~~ = , which will happen when 

^ do cos a 

and <f> are 90° and 270°, the corresponding positions of the driving 

a^m being ab 3 and ab 4 . 

Hence in one rotation of the driving shaft the velocity ratio varies 

twice between the limits and cos a ; and between these points there 

cos a: 

are four positions where the value is unity. 

If the angle a increases, the variation in the angular velocity ratio of 

the two connected shafts also increases ; and when this variation becomes 

too great to be admissible in any case, other arrangements must be 

employed. 



CHAPTER VIII. 



PARALLEL MOTIONS.— STRAIGHT-LINE MOTIONS. 



A parallel motion is a linkage designed to guide a reciprocating piece 
either exactly or approximately in a straight line, in order to avoid the 
friction arising from the use of straight guides. Some parallel motions 
are exact, that is, they guide the reciprocating piece in an exact straight 
line; others, which occur more frequently, are approximate, and are 
usually designed so that the middle and two extreme positions of the 
guided point shall be in one straight line, while at the same time care is 
taken that the intermediate positions deviate as little as possible from 
that line. 

126. Peaucellier's Straight-line Motion. — Fig. 173 shows a linkage, 
invented by M. Peaucellier, for describing an exact straight line within 
the limits of its motion. 

It consists of eight links joined at their ends. Four of these links, 

A, B, C, and D, 
are equal to each 
other and form: 
a cell; the two 
equal links E and 
F connect the 
opposite points 
of the cell a and 
e with the fixed 
centre of motion 
d; the link G = 
%bd oscillates on 
the fixed centre 




Fig. 173. 



c, cd thus forming the fixed link equal in length to G. 

If now the linkage be moved within the limits possible by its con- 
struction (that is, until the links B and G, and C and G come into line 
on opposite sides of the centre line of motion cd), the cell will open 

114 



PEAUCELLIERS STRAIGHT-LINE MOTION. 115 

and close; the points a and e will describe circular arcs about d, and 
b about c. Finally, the point p will describe a straight line ss perpen- 
dicular to the line of centres cd. 

To prove this, move the linkage into some other position, as p^^cd, 
(It is to be noticed that since the links A and C, B and D, and E and F T 
always form isosceles triangles with a common base, a straight line from 
p to d will always pass through b.) If the line traced by the point p is 
a straight line, the angle p x pd will be 90°. The angle bb x d is 90°, since 
bc = cd = b 1 c; therefore the triangles p t pd and bb x d would be similar 
right triangles, and we should have 

pd _b t d 
p x d bd 

To prove that ss is a straight line it is necessary to show that the 
Lbove relation exists in the different positions of the linkage. In Fig. 173 

E*=tf 2 +(bf+bdy ; 

B*=tf 2 +Ff;_ 

.: E 2 -B 2 = 2(bf)(bd) +bd 2 = bd(bd+2bf). 

But, since the links A and B are equal, the triangle pab is isosceles and 
the base pb = 2bf. 

... E 2 -B 2 =(bd)(pd) (44) 

By the same process, when the linkage is in any other position, as p x a}) x cd y 
we should have 

E x 2 -B 2 ={\d){V,d) (45) 

.Equatin equations (44) and (45) , 

(bd)(pd) = (bMv.d), . 
or 

pd b x d 

p x d bd ' 

which proves that the path of the point p is on the straight line ss. 

If the relation between the links cd and be be taken different from 
that shown (Fig. 173), the points b and p, sometimes called the poles 
of the cell, will be found to describe circular arcs whose centres are on 
the line passing through c and d ; in the case shown, one of these circu- 
lar arcs has a radius infinity. 

127. Scott Russell's Straight-line Motion. — This motion, suggested 
by Mr. Scott Russell, is an application of the isosceles sliding-block linkage f 
§ 117, shown in Fig. 144. 

It is made up of the links ab and pc, Fig. 174. The link ab, centred 




Fig. 174. 



110 PARALLEL MOTIONS.— STRAIGHT-LINE MOTIONS. 

at a, is joined to the middle point b of the link pc, and ah, be, and pb 

are taken equal to each other ; 
and the point c is constrained 
to move in the straight line 
ac by means of the sliding 
block. In this case the mo- 
tion of the sliding block c is 
slight, as the entire motion of p 
is seldom taken as great as cp. 
To show that the point p 
describes a straight line pp x p 2 
perpendicular to ac through 
a, a semicircle may be drawn 
through p and c with b as a 
centre ; it will also pass through a so that pac will be a right angle ; there- 
fore the point p is on ap, which is true for all positions of p. 

The point a should be located in the middle of the path or stroke of p. 
The motion of c may then be found by the equation 

a / 2 : — 2 

cc t = cp—v cp —ap , 
where ap is the half-stroke of p. 

Approximate straight-line motions somewhat resembling the preceding 
may be obtained by guiding the link cp entirely by oscillating links, 
instead of by a link and slide. 

1° In the link cp (Fig. 174) choose a convenient point e whose mean 
position is e v and whose extreme positions, are e and e 2 . Through these 
three points pass a circular arc, ee x e 2 , the centre of which / will be found 
on the line ac. Join e and / by a link ef, and the two links ab and ef 
will so guide pe that the mean and extreme positions of p will be found 
on the line pp 2 , provided suitable pairs are supplied to cause passage by 
the central position. 

2° The point c may be made to move very nearly in a straight line 
cc l by means of a link cd centred on a perpendicular erected at the 
middle point of the path of c. The longer this link the nearer the path 
of c will approach a straight line. 

This straight-line motion has been applied in a form of small sta- 
tionary engines, commonly known as grasshopper engines, where cbp 
(Fig. 174), extended beyond p, forms the beam of the engine, its right- 
hand end being supported by the link cd. The piston-rod is attached, 
by means of a cross-head, to the point p, which describes a straight line, 
and the connecting-rod is attached to a point in the line cp produced, 
both piston-rod and connecting-rod passing downward from cp. In 
this case it will be noticed that the pressure on the fulcrum c, of the 
beam, is equal to the difference of the pressures on the cross-head pin and 
ciank-pin instead of the sum, as in the ordinary form of beam engine. 



SCOTT RUSSELL'S STRAIGHT-LINE MOTION. 



117 



In this second form of motion it is not always convenient to place the 
point a in the line of motion pp 2 , and it is often located on one side, as 
shown in Fig. 175. 

The proportions of the different links which will cause the point p 
to be nearly on the 
straight line at the ex- 
treme positions and at 
the middle may be 
found as follows: 

Let pg be one half 
the stroke of the point 
p, and let the angle 
bac = 6, and bca = pbe = <f>. 
In this extreme position 
we may write 

ag = af-fg = af-be 
= ab cos d—pb cos 

= ab(l 




Fig. 175. 



■2 sin 2 



2 sin 



2 4> 



2 ,-p*(i--- 

But if the links are taken long enough, so that for a given stroke the 



angles 6 and <j> are small, then sin 
and 



0, nearly, and sin </> = <£, nearly, 



a, = a 6 (l-J)- P 5(l-|) 



= ab-pb- ab—+ ?&k 



If the linkage is now placed in its mid-position, 

ag = ab—pb. . . . 
Equating equations (46) and (47), 



(46) 
(47) 



or 



But in the triangle abc 



ab 2=P b f> 



a&_02 
pb~W' 



ab 

be sin 6 
ab ab' 
Pb~bc 



sin 6 d> 

= 7T> nearly; 



or 



(ab)(pb)=bc 



(48) 



(49) 



Hence the links must be so proportioned that be is a mean propor- 
tional between ab and pb, which also holds true when the path of p 
falls to the left of a instead of between a and c. 



118 



PARALLEL MOTION S. -STRAIGHT-LINE MOTIONS. 



As an example of the case where the path of the guided point falls 
to the left of a we have the straight-line motion of the Thompson steam- 
engine indicator, Fig. 181. 

128. Watt's Parallel Motion. — This motion is an application of the 
modified form of the double rocking lever (Fig. 109). 

Fig. 176 shows such a motion; here the two links ad and be con- 



nected by the link ab oscillate on 




Fig. 176. 



the fixed centres d and c, and any 
point, as p, in the connecting link 
ab will describe a complex curve. 
If the point p be properly chosen, 
a double-looped curve will be 
obtained, two parts of which 
are nearly straight lines. In de- 
signing such a motion it is cus- 
tomary to use only a portion ef of 
one of the approximate straight 
lines, and to so proportion the 
different links that the extreme 
and middle points e, f, and p shall 
be on a line perpendicular to the centre lines of the levers ad and be 
in their middle posi- 
tions, when they 
should be taken par- 
allel to each other. 

The linkage is 
shown in its mid-po- 
sition by dabc, Fig. 
177, and in the upper 
extreme position by 
dafiiC, where pp t is 
to be one half the 
stroke of p. Given 
the positions of the 
links ab and dc when 
in their mid-position, 
the axes c and d, the 
line of stroke ss, and 
the length of the 
stroke desired ; to 
find the points a and 
b, giving the link ab, 
and to prove that the point p, where ab crosses ss, will be found on the 
line ss when it is moved up (or down) one half the given stroke. Lay off 
on ss from the points g and h, where the links ad and be cross the line 




Fig. 177. 



WATT'S PARALLEL MOTION. 119 

ss, one quarter of the stroke, giving the points k and I; connect these 
points with the axes d and c respectively; draw the lines aka x and blb x 
perpendicular to dk and cl respectively, making aa x = 2ak and bb x = 2bl; 
then if the link centred at d were ad, it could swing to a x d, and similarly 
be could swing to b x c. By construction kg = ^ stroke, and aa 1 = 2ak; 
therefore a x e = \ stroke. Similarly bj = % stroke, which would make 
the figure ea x bj a parallelogram, and a x b x would equal ef. But ef is 
equal to ab, since bh = hf and ag = ge. Therefore if the linkage is dabc, 
it can occupy the position da x b x c; and since ap = ep = a x p x , and pp t = 
ea x =h the stroke, the point p will be at p x and ^ the stroke above p. 

To calculate the lengths of the links, given dg, eh, and gh, and the 
length of stroke S. Since the chord aa x of the arc through which a would 
move is bisected at right angles by the line dk, 

2 aS 2 

' gh ={ag)(dg)=—. 

S 2 s 2 

•'• a 9 = TaT~ an d ad=dg+ 



lQdg y ' lQdg' 

S 2 
Similarly bc = ch+ —- - ; 

ab = [gV+{ag+%hy]h 

To determine the position of the point p we have from the figure 

ap :bp=ag :bh (50) 

.\ ap : bp= TTT-r : ^ 7 r- r = ch : dg, 
y F Wdg 16ch ^/, 

from which 

ap : ab = ch : ch-\- dg, 

or bp : ab = dg : ch+dg, 

from which the position of the guided point p can be calculated. If, as 
is very often the case, ad = bc, then 

and bp :ab = dg : 2dg, 

or bp = ^ab, 

and the point p is thus at the middle of the link ab. 

This parallel motion may be arranged as shown in Fig. 178, where 
the centres c and d are on the same side of the line of motion. The 
graphical solution is the same as in Fig. 177, with the result that p is 
found where ab extended crosses the line of stroke ss, and, as before, it 
can be shown that if p is moved up one half the given stroke, it will 
be found on the line of stroke ss. 



120 



PARALLEL MOTIONS —STRAIGHT-LINE MOTIONS. 



In Fig. 177, letting the angle ada 1 =d and bcb 1 = <j), we have, from 




equation (50), 










ap 


ag ae 










bp 


bh bf 












ad( 1 — cos #) 


ad 2 


sin 


2 




bc(l — cos 


« 


be 2 


sin 


2 


which 


may be written 








ap be 
bp ad 


ad 

X — 

be* 


sur 
sin 2 



2 

i' 





2 

But ad sin 6 = bc sin <£ ; and since the 
angles or (f> would rarely exceed 20°, 

n t 

we may assume that ad sin — = be sin ^-. 



, nearly. 



(51) 



ap 6c 
op ad' 

or the segments of the link are in- 
versely proportional to the lengths of 
Fig. 178. fa e ne arer levers, which is the rule 

usually employed when the extreme positions can vary a very little from 
the straight line. When the levers are equal this rule is exact. 

129. The Pantograph. — The pantograph is a four-bar linkage so 
arranged as to form a paral- f 

lelogram abed, Fig. 179. 
Fixing some point in the 
linkage, as e, certain other 
points, as /, g, and h, will 
move parallel and similar to 
each other over any path 
either straight or curved. 
These points, as /, g, and h, 
must lie on the same straight 
line passing through the 
fixed point e, and their mo- 
tions will then be propor- 
tional to their distances from 
the fixed point. To prove 
that this is so, move the point / to any other position, as f x ; the linkage 
will then be found to occupy the position a 1 b 1 c 1 d v Connect f x with e; 




Fig. 179. 



THE PANTOGRAPH. 121 

then h lt where f x e crosses the link 6 1 c 1 , can be proved to be the same 

distance from q that h is from c, and the line hk x will be parallel to ff x * 

In the original position, since fd is parallel to he, we may write 

fd _de _fe 
he ce he' 

In the second position, since f x d x is parallel to \e x and since f x e is drawn 
a straight line, we have 

h x c x c x e h x e' 

Now in these equations — = — : therefore i- = T- 1 ', but fd = Ld*, 
^ ce c x e he h x c x ' 71 * 

which gives hc = h x c x , which proves that the point h has moved to h x . Also 

fe f e 

j- = t-> fro m which it follows that ff x is parallel to hh lf and 

ffi_ = f± = de 

hh x he ce ' 

or the motions are proportional to the distances of the points / and h 
from e. 

To connect two points, as a and b, Fig. 180, by a pantograph, so that 
their motions shall be parallel d 

and similar and in a given ratio, 
we have, first, that the fixed 
point c must be on the straight 
line ab continued, and so located 
that ac is to be as the desired 
ratio of the motion of a to b. 
After locating c, an infinite num- „ 

ber of pantographs might be 
drawn. Care must be taken that the links are so proportioned as to 
allow the desired magnitude and direction of motion. 

It is interesting to note that if b were the fixed point, a and c would 
move in opposite directions. It can be shown as before that their 
motions would be parallel and as ab is to be. 

The pantograph is often used to reduce or enlarge drawings, for it 
is evident that similar curves may be traced as well as straight lines. 
Also pantographs are used to increase or reduce motion in some definite 
proportion, as in the indicator rig on an engine where the motion of 
the cross-head is reduced proportionally to the desired length of the 
indicator diagram. When the points, as /and h (Fig. 179), are required 
to move in parallel straight lines it is not always necessary to employ 
a complete parallelogram, provided the mechanism is such that the 




122 



PARALLEL MOTIONS.— STRAIGHT-LINE MOTIONS. 



points / and h are properly guided. Such a case is shown in Fig. 181, 
which is a diagram of the mechanism for moving the pencil on a Thomp- 
son steam-engine indicator. The pencil at /, which traces the diagram 
on a paper carried by an oscillating drum, is guided by a Scott-Russell 
straight-line motion abed so that it moves nearly in a straight line ss 
parallel to the axis of the drum, and to the centre line of the cylinder 
tt. It must also be arranged that the motion of the pencil / always 
bears the same relation to the motion of the piston of the indicator 
on the line tt. To secure this draw a line from f to d and note the point 
e where it crosses the line tt: e will be a point on the piston-rod, which 




Fig. 181. 

rod is guided in an exact straight line by the cylinder. If now the link 
eh is added so that its centre line is parallel to cd, we should have, assum- 
ing / to move on an exact straight line, the motion of / parallel to the 
motion of e and in a constant ratio as cf : ch or as df : de. This can be seen 
by supposing the link eg to be added, which completes the pantograph 
dgehef. If eg were added, the link ab could not be used, as the linkage 
abedf does not give an exact straight-line motion to /. For constructive 
reasons the link eg is omitted; a ball joint is located at e which moves 
in an exact straight line, and the point / is guided by the Scott-Russell 
motion, the error in the motion being very slight indeed. 

Slides are often substituted, in the manner just explained, for links 
of a pantograph, and exact reductions are thereby obtained. In Fig. 182 
the points / and h are made to move on the parallel lines mm and nn 
respectively. Suppose it is desired to have the point h move J as much 
.as /. Draw the line fhe and lay off the point e so that eh:ef=l -3; draw 



APPLICATIONS OF WATT'S PARALLEL MOTION. 



123 



a line, as cd, and locate a point d upon it which when connected to / with 
a link df will move nearly an equal 
distance to the right and left of the 
line ef and above and below the line 
mm for the known motion of /. 
Draw ch through h and parallel to 
df. The linkage echdf will accom- 
plish the result required. The dotted 
link ah may be added to complete 
the pantograph, and the slide h may 
then be removed or not as desired. 
The figure also shows how a point g 
may be made to move in the oppo- 
site direction to / in the same ratio 
as h but on the line n^, the equivalent pantograph being drawn dotted. 
The link cd h shown in its extreme position to the left by heavy lines 
and to the right by light lines. 




130. Applications of Watt's Parallel Motion. — Watt's parallel 
motion has been much used in beam engines, and it is generally neces- 
sary to arrange so that more than one point can be guided, which is 
accomplished by a pantograph attachment. 

In Fig. 183 a parallel motion is arranged to guide three points p, p ly 

and p 2 in parallel straight 
lines. The case chosen is 
that of a compound con- 
densing beam engine, where 
P 2 is the piston-rod of the 
low-pressure cylinder, P 1 
that of the high-pressure 
cylinder, and P the pump- 
rod, all of which should 
move in parallel straight 
lines, perpendicular to the 
centre line of the beam in 
its middle position. 

The fundamental linkage 
dabc is arranged to guide the point p as required ; then adding the paral- 
lelograms astb and ap 2 rb, placing the links st and p 2 r so that they pass 
through the points p 1 and p 2 respectively, found by drawing the straight 
line cp and noting points p t and p 2 where it intersects lines P 1 and P 2 , 
we obtain the complete linkage. The links are arranged in two sets, 
and the rods are carried between them; the links da are also placed 
outside of the links p 2 a. When the point p falls within the beam a 




Fig. 183. 



124 PARALLEL MOTIONS.— STRAIGHT-LINE MOTIONS. 

double pump-rod must be used. The linkage is shown in its extreme 
upper position to render its construction clearer. 

The various links are usually designated as follows: cr the main 
beam, ad the radius-bar or bridle, p 2 r the main link, ab the back link, 
and p 2 a the parallel bar, connecting tne mam and back links. 

In order to proportion the linkage so that the point p 2 shall fall at 
the end of the link rp 2 we have, by similar triangles cbp and crp 2 , 

cb : bp = cr : rp 2 = cr : ab. 

abXcb 

.'. cr = — : . 

op 

The relative stroke £ of the point p 2 and s of the point p are expressed 
by the equation 

S:s = cp 2 :cp = cr: cb. 

If we denote by M and N the lengths of the perpendiculars dropped 
from c to the lines of motion P 2 and P respectively, then 

S:s = M:N 
and 

e M v N /AN 

S = s ¥ ; s^S^ (A) 

The problem will generally be, given the centres of the main beam 
c and bridle d, the stroke S of the point p 2 , and the paths of the guided 
points 2VPi> and p 2 , to find the remaining parts. The strokes of the 
guided points can be found from equation (A) and then the method 
of § 128, Fig. 177, can be applied. 

131. Roberts's Approximate Straight-line Motion. — This might 
also be called the W straight-line motion, and 
is shown in Fig. 184. It consists of a rigid 
triangular frame abp forming an isosceles 
triangle on ab, the points a and b being guided 
by links ad = bc = bp, oscillating on the centres 
d and c respectively, which are on the line of 
Fig. 184. motion dc. 

To lay out the motion, let dc be the straight 
line of the stroke along which the guided point p is to move approxi- 
mately, and p be the middle point of that line. Draw two equal isosceles 
triangles, dap and cbp; join ab, which must equal dp = pc. Then abp 
is the rigid triangular frame, p the guided point, and d and c are the 
centres of the two links. The extreme positions when p is at d and c 
are shown at da x a 2 and ca 2 b 2 , the point a 2 being common to both. The 
length of each side of the triangle, as ap = da, should not be less than 
1.186 dp, since in this case the points ca 2 a x and da 2 b 2 lie in straight lines. 



TCHEBICHEFF'S APPROXIMATE STRAIGHT-LINE MOTION. 125 



It may be made as much greater as the space will permit, and the greater 
it is the more accurate will the motion be. The intermediate positions 
between dp and cp vary somewhat from the line dc. 

132. Tchebicheff's Approximate Straight-line Motion. — Fig. 185 
shows another close approximation to a straight- 
line motion invented by Prof. Tchebicheff of St. 
Petersburg. It is an application of the double 
rocking lever (Fig. 109) with the levers crossed, 
cd being the fixed link. 

The links are made in the following proportion: 
If cd = 4:, then ac = bd = 5 and ab = 2. The guided 
point p is located midway between a and b on the 
link ab and is distant from cd an amount equal to V5 2 — 3 2 = 4. When 
the point p moves to p L , directly over d, dp 1 = db 1 — b 1 p 1 ^ 5 — 1=^4. Thus 
the middle and extreme positions of p, as shown, are in line, but the 
intermediate positions will be found to deviate slightly from the straight 
line. To render the range of motion shown on the figure possible the 
links ac and bd would need to be offset. 

133. Parallel Motion by Means of Four-bar Linkage. — The parallel 
crank mechanism, § 104, Fig. 105, is very often used to produce parallel 
motions. The common parallel ruler, consisting of two parallel straight- 
edges connected by two equal and parallelly-placed links is a familiar 





example of such application. A double parallel crank mechanism is 
applied in the Universal Drafting-machine, now extensively used in 
place of T square and triangles. Its essential features are shown in 



126 



PARALLEL MOTIONS.— STRAIGHT-LINE MOTIONS. 



Fig. 186. The clamp C is made fast to the upper left-hand edge of 
the drawing-board and supports the first linkage abdc. The ring cedf 
carries the second linkage efhg, guiding the head P. The two combined 
scales and straight-edges A and B, fixed at right angles to each other, 
are arranged to swivel on P, and by means of a graduated circle and 
clamp-nut may be set at any desired angle, the device thus serving as a 
protractor. The fine lines show how the linkages appear when the head 
is moved to P 1} and it is easily seen that the straight-edges will always 
be guided into parallel positions. 

134. Parallel Motion by Cords. — Cords, wire ropes, or small steel 
wires are frequently used to compel the motion of long narrow carriages 

or sliders into parallel positions. In 
Fig. 187 the slider R has at either end 
the double-grooved wheels E and F. 
A cord attached to the hook A passes 
vertically downward under F, across 
over E, and downward to the hook C. 
A similarly arranged cord starts from 
B, passes around E and F, using the 
remaining grooves, and is made fast 
to hook D. On moving the slider 
downward it will be seen that for a 
motion of 1" the wheel F will give out 
1" of the rope from A and take up 1" 
of rope from D, which is only possible 
when E takes up 1" from C and gives 
out 1" to B. Thus the slider R is constrained to move into parallel 





Fig. 188. 











^A s T BJ~ 






@H R ... E0 


1 




(°<-F " GV»i 


j 




W C (? 





Fig. 189. 



positions. In practice turnbuckles or other means are provided to keep 
the cords taut. 



PARALLEL MOTION BY CORDS. 



127 



Figs. 188 and 189 show two other arrangements which will accom- 
plish the same purpose. In Fig. 188, sometimes applied to guide 
straight-edges on drawing-boards, the cords or wires cross on the back 
side of the board where the four guide-wheels are located and the straight- 
edge R is guided by special fastenings E and F, passing around the edges 
of and under the board. By making one of these fastenings movable 
the straight-edge may be adjusted. Fig. 189 shows a similar device 
that might be applied on a drawing-board. Here the wires are on the 
front of the board and are arranged to pass under the straight-edge 
in a suitable groove. The turnbuckle T serves to keep the wires taut, 
and the slotted link S allows adjustment. 

The device shown in Fig. 187 is often known as a squaring-band and 
is applied in spinning-mules and in some forms of travelling cranes. 



CHAPTER IX. 

INTERMITTENT LINKWORK.— INTERMITTENT MOTION. 

A reciprocating motion in one piece may cause an intermittent 
circular or rectilinear motion in another piece. It may be arranged 
that one half of the reciprocating movement is suppressed and that the 
other half always produces motion in the same direction, giving the 
ratchet-wheel; or the reciprocating piece may act on opposite sides 
of a toothed wheel alternately, and allow the teeth to pass one at a 
time for each half reciprocation, giving the different forms of escape- 
ments as applied in timepieces. 

135. Ratchet-wheel. — A wheel, provided with suitably shaped 
pins or teeth, receiving an intermittent circular motion from some 
vibrating or reciprocating piece, is called a ratchet-wheel. 

In Fig. 190 A represents the ratchet-wheel turning upon the shaft 
a; C is an oscillating lever carrying the detent, click, or catch B, which 
acts on the teeth of the wheel. The whole forms the three-bar linkage 
acb. When the arm C moves left-handed, the click B will push the 
wheel A before it through a space dependent upon the motion of C. 
When the arm moves back, the click will slide over the points of the 
teeth, and will be ready to push the wheel on its forward motion as 
before; in any case, the click is held against the wheel either by its 
weight or the action of a spring. In order that the arm C may produce 
motion in the wheel A, its oscillation must be at least sufficient to cause 
the wheel to advance one tooth. 

It is often the case that the wheel A must be prevented from moving 
backward on the return of the click B. In such a case a fixed pawl, click, 
or detent, similar to B> turning on a fixed pin, is arranged to bear on the 
wheel, it being held in place by its weight or a spring. Fig. 190 might 
be taken to represent a retaining-pawl, in which case ac is a fixed link 
and the click B would prevent any right-handed motion of the wheel A. 
Fig. 191 shows a retaining-pawl which would prevent rotation of the 

128 



RATCHET-WHEEL. 



129 



wheel A in either direction ; such pawls are often used to retain pieces 
in definite adjusted positions. 

If the diameter of the wheel A (Fig. 190) be increased indefinitely, 
it will become a rack which would then receive an intermittent trans- 
lation on the vibration of the arm C : a retaining-pawl might be required 
in this case also to prevent a backward motion of the rack. 

A click may be arranged to push, as in Fig. 190 ; or to pull ; as in Fig. 




Fig. 190. 



Fig. 191, 



197. In order that a click or pawl may retain its hold on the tooth of a 
ratchet-wheel, the common normal to the acting surfaces of the click and 
tooth, or pawl and tooth, must pass inside of the axis of a pushing click 
or pawl, as shown on the lowest click, Fig. 192, and outside the axis of 
the pulling click or pawl; the normal might pass through the axis, but 
the pawl would be more securely held if the normal is located accord- 
ing to the above rule, which also secures the easy falling of the pawl 
over the points of the teeth. It is sometimes necessary, or more con- 
venient, to place the click-actuating lever on an axis different from that 
of the ratchet-wheel; in such a case care must be taken that in all 
positions of the click the common normal occupies the proper position; 
it will generally be sufficient to consider only the extreme positions of 
the pawl in any case. Since when the lever vibrates on the axis of the 
wheel, the common normal always makes the same angle with it in all 
positions, thus securing a good bearing of the pawl on the tooth, it is 
best to use this' construction when practicable. 

The effective stroke of a click or pawl is the space through which the 
ratchet-wheel is driven for each forward stroke of the arm. The total 
stroke of the arm should exceed the effective stroke by an amount 
sufficient to allow the click to fall freely into place. 

A common example of the application of the click and ratchet-wheel 
may be seen in several forms of ratchet-drills used to drill metals by 
hand. As examples of the retaining-pawl and wheel we have capstans 
and windlasses, where it is applied to prevent the recoil of the drum or 
barrel, for which purpose it is also applied in clocks. 




130 INTERMITTENT LINKWORK.— INTERMITTENT MOTION. 

It is sometimes desirable to hold a drum at shorter intervals than 
would correspond to the movement of one tooth of the ratchet-wheel; 

in such a case several equal pawls 
may be used. Fig. 192 shows a 
case where three pawls were used, 
all attached by pins c, c 1} c 2 to the 
fixed piece C, and so proportioned 
that they come into action alter- 
nately. Thus, when the wheel A 
has moved an amount corre- 
sponding to one-third of a tooth, 
the pawl B x will be in contact with 
the tooth 6 1 ; after the next one- 
third movement, B 2 will be in con- 
Fig. 192. tact with b 2 ; then after the remain- 

ing one- third movement, B will 
come into contact with the tooth under b; and so on. This arrangement 
enables us to obtain a slight motion and at the same time use compara- 
tively large and strong teeth on the wheel in place of small weak ones. 
The piece C might also be used as a driving arm, and the wheel could 
then be moved through a space less than that of a tooth. The three 
pawls might be made of different lengths and placed side by side on one 
pin, as c v in which case a wide wheel would be necessary: the number 
of pawls required would be fixed by the conditions in each case. 

136. Reversible Click or Pawl. — The usual form of the teeth of a 
ratchet-wheel is that given in Fig. 192, which only admits of motion in 
one direction; but in feed mechanisms, such as those in use on shapers 
and planers, it is often necessary to make use of a click and ratchet-wheel 
that will drive in either direction. Such an arrangement is shown 
in Fig. 193, where the wheel A has radial teeth, and the click, which is 
made symmetrical, can occupy either of the positions B or B f , thus giv- 
ing to A a right- or a left-handed motion. In order that the click B may 
be held firmly against the ratchet-wheel A in all positions of the arm C f 
its axis c, after passing through the arm, is provided with a small triangu- 
lar piece (shown dotted) ; this piece turning with B has a flat-ended 
presser, always urged upward by a spring (also shown dotted) bearing 
against the lower angle opposite B, thus urging the click toward the 
wheel; a similar action takes plac^ when the click is in the dotted posi- 
tion B' '. When the click is placed in line with the arm C, it is held in 
position by the side of the triangle parallel to the face of the click; thus 
this simple contrivance serves to hold the click so as to drive in either 
direction, and also to retain it in position when thrown out o£ 
gear. 



REVERSIBLE CLICK OR PAWL. 



131 



As for different classes of work a change in the "feed " is desired, we 
must arrange that the motion of the ratchet-wheel 
A (Fig. 193), which produces the feed, can be ad- 
justed. This is often done by changing the swing 
of the arm C, which is usually actuated by a rod 
attached at its free end. The other end of the rod 
is attached to a vibrating lever which has a definite 
angular movement at the proper time for the feed 
to occur, and is provided with a T slot in which 
the pivot for the rod can be adjusted by means of 
a thumb-screw and nut. By varying the distance 
of the nut from the centre of motion of the lever, 
the swing of the arm C can be regulated ; to reverse the 
feed, it occurring in the same position as before, the click must be reversed 
and the nut moved to the other side of the centre of swing of the lever. 

Figs. 194 and 195 show other methods of adjusting the motion of the 
ratchet-wheel. In Fig. 194, which shows a form of feed mechanism 
used by Sir J. Whitworth in his planing-machines, C is an arm carrying 
the click B, and swinging loosely on the shaft a fixed to the ratchet- 
wheel A. The wheel E, also turning loosely on the shaft a, and placed 
just behind the arm C, has a definite angular motion sufficient to produce 




Fig. 193. 




Fig. 194. Fig. 195. 

the coarsest feed desired; its concentric slot m is provided with two 
adjustable pins ee, held in place by nuts at their back ends, and enclosing 
the lever C, but not of sufficient length to reach the click B. When the 
pins are placed at the ends of the slot, no motion will occur in the arm C*, 
but when e and e are placed as near as possible to each other, confining 
the arm C between them, all of the motion of E will be given to the arm 
C, thus producing the greatest feed ; any other positions of the pins will 
give motions between the above limits, and the adjustment may be made 
to suit each case. 

Fig. 195 shows another method of adjusting the motion of the ratchet- 
wheel A. The stationary shaft a, made fast to the frame of the machine 
at m, carries the vibrating arm C, ratchet-wheel A, and adjustable shield 
S; the two former turn loosely on the shaft, while the latter is made fast 
to it by means of a nut n, the hole in S be ng made smaller than that in 



132 INTERMITTENT LINKWORK.— INTERMITTENT MOTION. 

A, to provide a shoulder against which S is held by the nut. The arm 
C carries a pawl B of a thickness equal to that of the wheel plus that of 
the shield S; the extreme positions of this pawl are shown by dotted 
lines at B' and B" . The teeth of the wheel A may be made of such 
shape as to gear with another wheel operating the feed mechanism; or 
another wheel, gearing with the feed mechanism, might be made fast to 
the back of A, if more convenient: in the latter case, the arm C would be 
placed back of this second wheel. 

If we suppose the lever in its extreme left position, the click will be 
at B" resting upon the face of the shield S, which projects beyond the 
points of the teeth of A ; and in the right-handed motion of the lever the 
click will be carried by the shield S until it reaches the position B, where 
it will leave the shield and come in contact with the tooth b, which it will 
push to V in the remainder of the swing. In the backward swing of the 
lever the click will be drawn over the teeth of the wheel and face of the 
shield to the position B" . In the position of the shield shown in the 
figure a feed corresponding to three teeth of the wheel A is produced ; by 
turning the shield to the left one, two, or three teeth, a feed of four, five, 
or six teeth might be obtained ; while, by turning it to the right, the feed 
could be diminished, the shield S being usually made large enough to 
consume the entire swing of the arm C. This form of feed mechanism 
is often used in slotting-machines, and in such cases, as well as in Figs. 
194 and 195, the click is usually held to its work by gravity. 

137. Double-acting Click. — This device consists of two clicks mak- 
ing alternate strokes, so as to produce a nearly continuous motion of the 
ratchet-wheel which they drive, that motion being intermittent only at the 
instant of reversal of the movement of the clicks. In Fig. 196 the clicks act 




Fig. 196. 

by pushing, and in Fig. 197 by pulling ; the former arrangement is generally 
best adapted to cases where much strength is required, as in windlasses. 
Each single stroke of the click-arms cdc' (Fig. 196) advances the 
ratchet-wheel through one-half of its pitch or some multiple of its half- 
pitch. To make this evident, suppose that the double click is to advance 
the ratchet-wheel one tooth for each double stroke of the click-arms, 



DOUBLE-ACTING CLICK. 133 

the arms being shown in their mid-stroke position in the figure. Now 
when the click be is beginning its forward stroke, the click b'c' has just 
completed its forward stroke and is begining its backward stroke ; during 
the forward stroke of be the ratchet-wheel will be advanced one-half 
a tooth; the click b'c' , being at the same time drawn back one-half 
a tooth, will fall into position ready to drive its tooth in the remaining 
single stroke of the click-arms, which are made equal in length. By 
the same reasoning it may be seen that the wheel can be moved ahead 
some whole number of teeth for each double stroke of the click-arms. 

In Fig. 196 let the axis a and dimensions of the ratchet-wheel be 
given, also its pitch circle BB, which is located half-way between the 
tips and roots of the teeth. Draw any convenient radius ab, and from it 
lay off the angle bae equal to the mean obliquity of action of the clicks, 
that is, the angle that the lines of action of the clicks at mid-stroke are 
to make with the tangent to the pitch circle through the points of action. 
On ae let fall the perpendicular be, and with the radius ae describe the 
circle CC: this is the base circle, to which the lines of action of the clicks 
should be tangent. Lay off the angle eaf equal to an odd number of times 
the half-pitch angle, and through the points e and /, on the base circle, 
draw two tangents cutting each other in h. Draw hd bisecting the angle 
at h, and choose any convenient point in it, as d, for the centre of the 
rocking shaft, to carry the click-arms. From d let fall the perpendiculars 
dc and dc' on the tangents hec and fhc' respectively; then c and c' will be 
the positions of the click-pins, and dc and dc' the centre lines of the 
click-arms at mid-stroke. Let b and b' be the points where ce and c'f cut 
the pitch circle; then cb and c'b' will be the lengths of the clicks. The 
effective stroke of each click will be equal to half the pitch as measured 
on the base circle CC (or some whole number of times this half-pitch), 
and ihe total stroke must be enough greater to make the clicks clear 
the teeth and drop well into place. 

In Fig. 197 the clicks pull instead of push, the obliquity of action is 
zero, and the base circle and pitch 
circle become one, the points b, 
e, and b' , f (Fig. ,196) becoming e 
and / (Fig. 197). In all other 
respects the construction is the 
same as when the clicks act 
by pushing, and the different 
points are lettered the same as in 
Fig. 196. 

Since springs are liable to lose 
their elasticity or become broken 
after being in use some time, it is 
often desirable to get along without applying them to keep clicks in position. 




Fig. 197. 



134 INTERMITTENT LINKWORK.— INTERMITTENT MOTION. 

Fig. 198 shows in elevation a mechanism where no springs are required 
to keep the clicks in place, it being used in some forms of lawn-mowers to 
connect the wheels to the revolving cutter when the mower is pushed 
forward, and to allow a free backward motion of the mower while the 
cutter still revolves. The ratchet A is usually made on the inside of the 
wheels carrying the mower, and the piece C, turning on the same axis as A, 
carries the three equidistant pawls or clicks B, shaped to move in the 
cavities provided for them. In any position of C, at least one of the 
clicks will be held in contact with A by the action of gravity, and any 
motion of A in the direction of the arrow will be given to the piece C. 
Here the ratchet-wheel drives the click, ac being the actuated click-lever. 




Fig. 198. Fig. 199. 

The piece C is sometimes attached to a roller by means of the shaft a; 
then any left-handed motion of C will be given to A, while the right- 
handed motion will simply cause the clicks to slide over the teeth of A. 
The clicks B are usually held in place by a cap attached to C. 

Fig. 199 shows a form of click which is always thrown into action 
when a left-handed rotation is given to its arm C, while any motion of 
the wheel A left-handed will immediately throw the click out of action. 
The wheel A carries a projecting hub d, over which a spring D is fitted 
so as to move with slight friction. One end of this spring passes between 
two pins, e, placed upon an arm attached to the click B. When the 
arm C is turned left-handed, the wheel A and the spring D being sta- 
tionary, the click B will be thrown toward the wheel by the action of 
the spring on the pin e. The motion of the wheel A will be equal to 
that of the arm C, minus the motion of C necessary to throw the click 
into gear. Similarly, when A turns left-handed, the click B is thrown 
out of gear. This mechanism is employed in some forms of spinning- 
mules to actuate the spindles when winding on the spun yarn. 

138. Friction-catch. — Various forms of catches depending upon 
friction are often used in place of clicks; these catches usually act upon 
the face of the wheel or in a suitably formed groove cut in the face. 



FRICTION-CATCH. 



135 



Friction-catches have the advantage of being noiseless and allowing any 
motion of the wheel, as they can take hold at any point; they have the 
disadvantage, however, of slipping when worn, and of getting out of 
order. 

Fig. 200 shows a friction-catch B working in a V-shaped groove in 
the wheel A, as shown in section A'B' . Here B acts as a retaining- 
click, and prevents any right-handed motion of A ; its face is circular in 
outline, the centre being located at d, a little above the axis c. A simi- 
larly shaped catch might be used in place of an actuating click to cause 
motion of A. 

Fig. 201 shows four catches like B (Fig. 200) applied to drive an 
annular ring A in the direction indicated by the arrow. When the 
piece c is turned right-handed, the catches B are thrown against the 




Fig. 200. 



Fig. 201. 



inside b of the annular ring by means of the four springs shown; on 
stopping the motion of c, the pieces B are pushed, by the action of b, 
toward the springs which slightly press them against the ring and 
hold them in readiness to again grip when c moves right-handed. Thus 
an oscillation of the piece c might cause continuous rotation of the 
wheel A, provided a fly-wheel were applied to A to keep it going while 
c was being moved back. The annular ring A is fast to a disc carried 
by the shaft a; the piece c turning loosely on a has a collar to keep it 
in position lengthwise of the shaft. 

The nipping-lever shown in Fig. 202 is another application of the 
friction-catch. A loose ring C surrounds the wheel A ; a friction-catch 
B having a hollow face works in a pocket in the ring and is pivoted 
at c. On applying a force at the end of the catch B in the direction 
of the arrow, the hollow face of the catch will "nip" the wheel at b, 
and cause the ring to bear tightly against the left-hand part of the cir- 
cumference of the wheel; the friction thus set up will cause the catch, 
ring, and wheel to move together as one piece. The greater the pull 
applied at the end of the catch the greater will be the friction, as the 



136 INTERMITTENT LINKWORK.— INTERMITTENT MOTION. 



friction is proportional to the pressure; thus the amount of friction, 
developed will depend upon the resistance to motion of A. Upon 
reversing the force at the end of the catch, the hollow face of the catch 
will be drawn away from the face of A, and the rounding top part 
of the catch, coming in contact with the top of the cavity in the ring, 
will cause the ring to slide back upon the disc. An upward motion of 
the click end will again cause the wheel A to move forward, and thus 
the action is the same as in a ratchet and wheel. 





Fig. 202. 



Fig. 203. 



Fig. 203 shows, in section, a device which has been applied to actu- 
ate sewing-machines in place of the common crank. Two such mechan- 
isms were used, one to rotate the shaft of the machine on a downward 
tip of the treadle, while the other acted during the upward tip, the 
treadle-rods being attached to the projections of the pieces B. The 
mechanism shown in the figure acts upon the shaft during the down- 
ward motion of the projection B as shown by the arrow. 

The piece C, containing an annular groove, is made fast to the 
shaft a, the sides of this groove being turned circular and concentric 
with the shaft. The piece B, having a projecting hub fitting loosely on 
the inner surface of the groove in C, is placed over the open groove, 
and is held in place by a collar on the shaft. The hub on the piece B, 
and the piece C, are shown in section. The friction-catch D, working in 
the groove, is fitted over the hub of B, the hole in D being elongated in 
the direction ab so that D can move slightly upon the hub and between 
the two pins e fast in the piece B. A cylindrical roller c is placed in 
the wedge-shaped space between the outer side of the groove and the 
piece D, a spring always actuating this roller in a direction opposite to 
that of the arrow, or towards the narrower part of the space. 

Now when the piece B is turned in the direction of the arrow by a 
downward stroke of the treadle-rod, it will move the piece D with it by 
means of the pins e; at the same time the roller c will move into the 
narrow part of the wedge-shaped space between C and D, and cause 
binding between the pieces D and C at b and at the surface of the roller. 
The friction at b thus set up will cause the motion of D to be given to 
C. On the upward motion of the projection B the roller will be moved 




FRICTION-CATCH. 137 

to the large part of its space by the action of the piece C revolving with 
the shaft combined with that of the backward movement of D, thus re- 
leasing the pressure at b and allowing C to move freely onward. The 
other catch would be made just the reverse of this one, and would act 
on an upward movement of the treadle-rod. 

Another form of friction catch, sometimes used in gang saws to 
secure the advance of the timber for each stroke of the saw, and called 
the silent feed, is shown in Fig. 204. 

The saddle-block B, which rests upon 
the outer rim of the annular wheel A, 
carries the lever C turning upon the pin 
c. The block D, which fits the inner 
rim of the wheel, is carried by the lever 
C, and is securely held to its lower end 
by the pin d on which D can freely turn. 
When the pieces occupy the positions 
shown in the figure, a small space exists 
between the piece D and the inside of the 
rim A. 

The upper end of the lever C has a reciprocating motion imparted to 
it by means of the rod E. The oscillation of the lever about the pin c is 
limited by the stops e and G carried by the saddle-block B. When the 
rod E is moved in the direction indicated by the arrow, the lever turning 
on c will cause the block D to approach B, and thus nip the rim at a and 
b; and any further motion of C will be given to the wheel A. On mov- 
ing E in the opposite direction the grip will first be loosened, and the 
lever striking against the stop e will cause the combination to slide freely 
back on the rim A. The amount of movement given to the wheel can be 
regulated by changing the stroke of the rod E by an arrangement similar 
to that described in connection with the reversible click, § 136. The 
stop G can be adjusted by means of the screw F so as to prevent the 
oscillation of the lever upon its centre c, thus throwing the grip out of 
action. The saddle-block B then merely slides back and forth on the 
rim, the action being the same as that obtained by throwing the ordinary 
click out of gear. 

139. Masked Wheels. — It is sometimes required that certain strokes 
of the click-actuating lever shall remain inoperative upon the ratchet- 
wheel. Such arrangements are made use of in numbering-machines 
where it is desired to print the same number twice in succession; they 
are called masked wheels. 

Fig. 205, taken from a model, illustrates the action of a masked 
wheel ; the pin-wheel D represents the first ratchet-wheel, and is fast to 
the axis a; the second wheel A has its teeth arranged in pairs, every 
alternate tooth being cut deeper, and it turns loosely on the axis a. 




138 INTERMITTENT LINKWORK.— INTERMITTENT MOTION. 

The click B is so made that one of its acting surfaces, i, bears against 
the pins e of the wheel D, while the other, g, is 
placed so as to clear the pins and yet bear upon 
the teeth of A, the wheel A being located so as 
to admit of this. 

If now we suppose the lever C to vibrate 

through an angle sufficient to move either wheel 

along one tooth, both having the same number, 

it will be noticed that when the projecting piece 

g is resting in a shallow tooth of the wheel A, 

the acting surface i will be retained too far from 

the axis to act upon the tooth e, and thus this 

vibration of the lever will have no effect upon 

the pin-wheel D, while when the piece g rests in a deep tooth, as b' ', the 

click will be allowed to drop so as to bring the surface i into action with 

the pin e f . 

In the figure the click B has just pushed the tooth e' into its present 
position, the projection g having rested in the deep tooth b r of the wheel 
A ; on moving back, g has slipped into the shallow tooth b, and thus the 
next stroke of the lever and click will remain inoperative on the wheel 
D, which advances but one tooth for every two complete oscillations of 
the lever C. 

Both wheels should be provided with retaining-pawls, one of which, 
7), is shown. This form of pawl, consisting of a roller p turning about 
an axis carried by the spring s, attached to the frame carrying the mech- 
anism, is often used in connection with pin-wheels, as by rolling between 
the teeth it always retains them in the same position relative to the axis 
of the roller; a triangular-pointed pawl which also passes between the 
pins is sometimes used in place of the roller. 

The pins of the wheel D might be replaced by teeth so made that 
their points would be just inside of the bottoms of the shallow teeth of 
A; a wide pawl would then be used, and when it rested in a shallow 
tooth of A it would remain inoperative on D, while when it rested in a 
deep tooth it would come in contact with the adjacent tooth of D and 
push it along. 

So long as the click B and the wheels have the proper relative motion 
it makes no difference which we consider as fixed, as the action will be 
the same whether we consider the axis of the wheels as fixed and the 
click to move, or the click to be fixed and the axis to have the proper 
relative motion in regard to it. The latter method is made use of in 
some forms of numbering-machines. 

140. Fig. 206 shows the mechanism of a " counter" used to record 
the number of double strokes made by a pump; the revolutions made 
by a steam engine, paddle, propeller, or other shaft, etc. Two views 



MASKED WHEELS. 



139 



are given in the figure, which represents a counter capable of recording 
revolutions from 1 to 999; if it is desired to record higher numbers, it 
will only be necessary to add more wheels, such as A. A plate, having 
a long slot or series of openings opposite the figures 000, is placed over 
the wheels, thus only allowing the numbers to be visible as they come 
under the sloe or openings. 

The number wheels A, A v A 2 , are arranged to turn loosely side by 
side upon the small shaft a, and are provided with a series of ten teeth cut 
into one side of their faces, while upon the other side a single notch is 
€ut opposite the zero tooth on the first side, it having the same depth 
and contour. This single notch can be omitted on the last wheel A 2 . The 
numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, are printed upon the faces of the 
wheels in proper relative positions to the teeth t. 

Two arms C are arranged to vibrate upon the shaft a of the number 
wheels, and carry at their outer ends the pin c, on which a series of 
clicks, b, b v and b 2 , are arranged, collars placed between them serving to 
keep them in position on the pin. The arms are made to vibrate through 
an angle sufficient to advance the wheels one tooth, i.e., one-tenth of a 
turn; their position after advancing a tooth is shown by dotted lines 
in the side view. A common method of obtaining this vibration is 
to attach a rod at r, one end of the pin c, this rod to be so attached at 
its other end to the machine as to cause the required backward and 
forward vibrations of the lever C for each double stroke or revolution 
that the counter is to record. 

The click b is narrow, and works upon the toothed edge of the first 
w r heel A, advancing it one tooth for every double stroke of the arm c. The 
remaining clicks b ± and b 2 are made broad, and work 
on the toothed edges of A x and A 2 , as well as on the 
notched rims of A and A v respectively. When the 
notches n and m come under the clicks 6 X and b 2 the 
clicks will be allowed to fall and act on the toothed 
parts of A x and A 2 ; but in any other positions of the 
notches the clicks will remain inoperative upon the 
wheels, simply riding upon the smooth rims of A and 
A t , which keep the clicks out of action. Each wheel 
is provided with a retaining-spring s to keep it in 
proper position. 

Having placed the wheels in the position shown 
in the figure, the reading being 000, the action is as 
follows : The click b moves the wheel A along one 
tooth for each double stroke of the arm C, the clicks 6 X and b 2 remain- 
ing inoperative on A t and A 2 ; on the figure 9 reaching the slot, or 
the position now occupied by 0, the notch n will allow the click 6 X 
to fall into the tooth 1 of the wheel A x , and the next forward stroke of 




Fig. 206. 



140 INTERMITTENT LINKWORK,— INTERMITTENT MOTION. 

the arm will advance both the wheels A and A v giving the reading 
10; the notch n having now moved along, the click b t will remain 
inoperative until the reading is 19, when 6 t will again come into action 
and advance A x one tooth, giving the reading 20; and so on up to 90, 
when the notch m comes under the click b 2 . To prevent the click b 2 
from acting on the next forward stroke of the arm, which would make the 
reading 101 instead of 91, as it should be, a small strip i is fastened firmly 
to the end of the click b 2 , its free end resting upon the click b v This 
strip prevents the click b 2 from acting until the click b x falls, which occurs 
when the reading is 99; on the next forward stroke the clicks b L and b 2 
act, thus giving the reading 100. As the strip merely rests upon b v it 
cannot prevent its action at any time. If another wheel were added, 
its click would require a strip resting on the end of b 2 . A substitute 
for these strips might be obtained by making the wheel A fast to the 
shaft a, and allowing the remaining wheels to turn loose upon it, thin 
discs, having the same contour as the notched edge of the wheel A , being 
placed between the wheels A ± A 2 , A 2 A 3 , etc., and made fast to the shaft, 
the notches all being placed opposite n; thus the edges of the discs 
would keep the clicks b 2 , b 3 , etc., out of action, except when the figure 
9 of the wheel A is opposite the slot, and the notches m, etc., are in proper 
position. A simpler form of counter will be described in § 141. 



Intermittent Motion. 

So far we have considered a uniform reciprocating motion in one 
piece, as giving an intermittent circular or rectilinear motion to another, 
the click being the driver and the wheel the follower. 

141. It is often required that a uniform circular motion of the driver 
shall produce an intermittent circular or rectilinear motion of the 
follower. The following examples will give some solutions of the prob- 
lem : 

Fig. 207 shows a combination by which the toothed wheel A is 

moved in the direction of the arrow, 
one tooth for every complete turn of 
the shaft d, the pawl B retaining the 
wheel in position when the tooth t on 
the shaft d is out of action. The sta- 
tionary link adc forms the frame, and 
provides bearings for the shafts d and 
a, and a pin c for the pawl B. The 
Fig. 207. arm e, placed by the side of the tooth 

upon the shaft, is arranged to clear 
the wheel A in its motion, and to lift the pawl B at the time when the 
tooth t comes into action with the wheel, and to drop the pawl when the 




GENEVA STOP. 



141 




Fig. 208. 



action of t ceases, i.e., when the wheel has been advanced one tooth. 
This is accomplished by attaching the piece n to the pawl, its contour 
in the raised position of the pawl being an arc of a circle about the centre 
of the shaft d; its length is arranged to suit the above requirements. 
When the tooth t comes in contact w T ith the wheel, the arm e, striking the 
piece n, raises the pawl (which is held in position by the spring s), and 
retains it in the raised position until the tooth t is ready to leave the 
wheel, when e, passing off from the end of n, allows the pawl to 
drop. 

In Fig. 208 the wheel A makes one- third of a revolution for every 
turn of the wheel be, its period of rest being about one- 
half the period of revolution of be. If we suppose A the 
follower, and to turn right-handed while the driver be 
turns left-handed, one of the round pins b is just about 
to push ahead the long tooth of A, the circular retaining 
sector c being in such a position as to follow a right-handed 
motion of A. The first pin slides down the long tooth, 
and the other pins pass into and gear with the teeth b' ', 
the last pin passing off on the long tooth e, when the 
sector c will come in contact with the arc c', and retain the wheel A 
until the wheel be again reaches its present position. 

Geneva Stop. — In Fig. 209 the wheel A makes one-sixth of a revo- 
lution for one turn of the driver ac, the pin b working 
in the slots b' causing the motion of A ; while the circular 
portion c of the driver, coming in contact with the cor- 
responding circular hollows e f , retains A in position when 
the tooth b is out of action. The wheel a is cut away 
just back of the pin b to provide clearance for the wheel 
A in its motion. If we close up one of the slots, as b', 
it will be found that the shaft a can only make a little 
over five and one-half revolutions in either direction before 
the pin b will strike the closed slot. This mechanism, when so modi- 
fied, has been applied to watches to prevent overwinding, and is called 
the Geneva stop, the wheel a being attached to the spring-shaft so as to 
turn with it, while A turns on an axis d in the spring-barrel. The num- 
ber of slots in A depends upon the number of times it is desired to turn 
the spring-shaft. 

By placing another pin opposite b in the wheel ac, as shown by 
dotted lines, and providing the necessary clearance, the wheel A could 
be moved through one-sixth of a turn for every half turn of ac. 

A simple type of counter extensively used on water-meters is shown 
in Fig. 210. It consists of a series of wheels A, B, C, mounted side by 
side and turning loosely on the shaft S; or the first wheel to right may 
be fast to the shaft and all the remaining wheels loose upon it. Each 




Fig. 209. 



142 INTERMITTENT LINKWORK.— INTERMITTENT MOTION. 

wheel is numbered on its face as in Fig. 206, and it is provided, as shown, 
that the middle row of figures appears in a suitable slot in the face of 
the counter. The first wheel A is attached to the worm-wheel E, hav- 
ing 20 teeth and driven by the worm F geared to turn twice for one 
turn of the counter driving shaft. 

On a parallel shaft T loose pinions D are arranged between each 
pair of wheels. Each pinion is supplied with six teeth on its left side 
extending over a little more than one-half its face and with three teeth, 
each alternate tooth being cut away, for the remainder of the face, as 
clearly shown in the sectional elevations. The middle elevation (Fig. 
210) shows a view of the wheel B from the right of the line ab with the 
pinion D sectioned on the line cd. The right elevation shows a view of 




Fig. 210. 



the wheel A from the left of the line ab with the pinion D sectioned on 
the line cd. The first wheel A, and all others except the last, at the left, 
have on their left sides a double tooth G, which is arranged to come in 
contact with the six-tooth portion of the pinion; the space between 
these teeth is extended through the brass plate, whose periphery H acts 
as a stop for the three-tooth portion of the pinion, as clearly shown in 
the figure to the right. Similarly on the right side of each wheel, except 
the first, is placed a wheel of 20 teeth gearing with the six-tooth part 
of the pinion, as shown in the middle figure. When the digit 9 on any 
wheel, except the one at the left, comes under the slot, the double tooth 
G is ready to come in contact with the pinion ; as the digit 9 passes under 
the slot the tooth G starts the pinion, which is then free to make one- 
third of a turn and again become locked by the periphery H. Thus any 
wheel to the left receives one-tenth of a turn for every passage of the 
digit 9 on the wheel to its right. In the figure the reading 329 will 
change to 330 on the passing of the digit 9. This counter can be made 
to record oscillations by supplying its actuating shaft with a ten-tooth 



INTERMITTENT MOTION. 



143 




Fig. 211b. 



ratchet, arranged with a click to move one tooth for each double oscilla- 
tion. 

Figs. 211a and 211b show two methods of advancing the wheels A 
through a space corresponding to one tooth during a small part of a 
revolution of the shafts c; 
in this case the shafts are at 
right angles to each other. 
In Fig. 211a a raised circu- 
lar ring with a small spiral 
part b attached to a disc is 
made use of; the circular 
part of the ring retains the 
wheel in position, while the spiral part gives it its motion. In Fig. 211b 
the disc carried by the shaft cc has a part of its edge bent helically at 
b; this helical part gives motion to the wheel, and the remaining 
part of the disc edge retains the wheel in position. By using a 

regular spiral, in Fig. 211a, 
and one turn of a helix, in 
Fig. 211b, the wheels A 
could be made to move uni- 
formly through the space of 
one tooth during a uniform 

) revolution of the shafts c. 
In Fig. 212 the wheel A 
is arranged to turn the wheel 
B, on a shaft at right angles 
to that of A, through one- 
half a turn while it turns 
one-sixth of a turn, and to 
lock B during the remaining 
five-sixths of the turn. 

The Star Wheel. — In/ 
Fig. 213 the wheel A, 
commonly known as the 
star wheel, turns through a space corresponding to one tooth for each 
revolution of the arm carrying the pin b and turning on 
the shaft c. The pin b is often stationary, and the star 
wheel is moved past it; the action is then evidently 
the same, as the pin and wheel have the same relative 
motion in regard to each other during the time of 
action. The star wheel is often used on moving parts of 
machines to actuate some feed mechanism, as may be seen Ftg - 213 - 
in cylinder-boring machines on the facing attachment, and in spinning- 
machinery. 




144 INTERMITTENT LINKWORK.— INTERMITTENT MOTION. 



142. Cam and Slotted Sliding Bar. — Fig. 214 shows an equilateral 
triangle abc, formed by three circular arcs, whose centres are at a, b, 
and c, the whole turning about the axis a, and producing an intermittent 
motion in the slotted piece B. The width of the slot is equal to the 
radius of the three circular arcs composing the three equal sides of the 
triangular cam A, and therefore the cam will always bear against both 
sides of the groove. 

If we imagine the cam to start from the position shown in Fig. 215 
when b is at 1, the slotted piece B will remain at rest while b moves 
from 1 to 2 (one-sixth of the circle 1, 2 ... 6), the cam edge be merely 
sliding over the lower side of the slot. When b moves from 2 to 3, i.e., 
from the position of A, shown by light full lines, to that shown by 
dotted lines, the edge ab will act upon the upper side of the slot, and 
impart to B a motion similar to that obtained in Fig. 156, being that of 
a crank with an infinite connecting-rod ; from 3 to 4 the point b will drive 
the upper side of the slot, ca sliding over the lower side, the motion here 
being also that of a connecting-rod with an infinite link, but decreasing 

instead of increasing as from 
2 to 3. When b moves from 
4 to 5 there is no motion in B ; 
from 5 to 6, c acts upon the 
upper side of the slot, and B 
moves downward; from 6 to 
1, ac acts on the upper side of 
the slot, and B moves down- 
ward to its starting position. 
The motion of B is accelerated 
from 5 to 6 and retarded from 
6 to 1. 




Fig. 214. 



Fig. 215. 

At A' a form of cam is shown where the shaft a is wholly contained 
in the cam. In this case draw the arcs de and cb from the axis of the 
shaft as a centre, making ce equal to the width of the slot in B; from c 
as a centre with a radius ce draw the arc eb, and note the point b where 
it cuts the arc cb; with the same radius and b as a centre draw the arc 
dc, which will complete the cam. In this case the angle cab will not be 
equal to 60°, and the motions in their durations and extent will vary a 
little from those described above. 

Locking Devices.— The principle of the slotted sliding bar combined 
with that of the Geneva stop is applied in the shipper mechanism shown 
in Fig. 216, often used on machines where the motion is automatically 
reversed. The shipper bar B slides in the piece CC, which also pro- 
vides a pivot a for the weighted lever wab. The end of the lever b 
opposite the weight w carries a pin which works in the grooved lug s 
on the shipper bar. In the present position of the pieces, the pin b is 




LOCKING DEVICES. 145 

in the upper part of the slot, and the weight w, tending to fall under 
the action of gravity, holds it there, the shipper being thus effectually 
locked in its present position. If now the 
lever be turned left-handed about its axis a 
until the weight w is just a little to the left 
of a, gravity will carry the weight and lever 
into the dotted position shown, where it will 
be locked until the lever is turned right- -p IG 2 i6 

handed. The principle of using a weight to 

complete the motion is very convenient, as the part of the machine actuat- 
ing the shipper often stops before the belt is carried to the wheel which 
produces the reverse motion, and the machine is thus stopped. The 
motion can always be made sufficient to raise a weighted lever, as shown 
above, and the weight will, in falling, complete the motion of the shipper. 
The device shown in Fig. 217, of which we may have many forms, 
serves to retain a wheel A in definite adjusted positions, its use being 
the same as that of the retaining-pawl shown in Fig. 191. The wheels 
B and A turn on the shafts c and a, respectively, carried by the link C, 
which is shown dotted, as it has been cut away in taking the section. 
Two positions of the wheel B will allow the teeth b of A to pass freely 
through its slotted opening, while any other position effectually locks 
the wheel A . The shape of the slot in B and the teeth of A are clearly 
shown in the figure. 





Fig. 217. Fig. 218. 

Fig. 218 shows another device for locking the wheel A, the teeth of 
which are round pins; but in this case it is necessary to turn B once to 
pass a tooth of A. If we suppose the wheel A under the influence of a 
spring which tends to turn it right-handed, and then turn B uniformly 
either right- or left-handed, the wheel A will advance one tooth for each 
complete turn of B, a pin first slipping into the groove on the left and 
leaving it when the groove opens toward the right, the next pin then 
coming against the circular part of B opposite the groove. It will be 



146 INTERMITTENT LINKWORK.— INTERMITTENT MOTION. 

noticed that while there are only six pins on the wheel A, yet there are 
twelve positions in which A can be locked, as a tooth may be in the 
bottom of the groove or two teeth may be bearing against the circular 
outside of B. Devices similar in principle to those shown in Figs. 217 
and 218 are often used to adjust stops in connection with feed 
mechanisms. 

Clicks and pawls as used in practice may have many different forms 
and arrangements; their shape depends very much upon their strength 
and the space in which they are to be placed, and the arrangement 
depends on the requirements in each case. 

Escapements. 

143. An escapement is a combination in which a toothed wheel acts 
upon two distinct pieces or pallets attached to a reciprocating frame, it 
being so arranged that when one tooth escapes or ceases to drive its 
pallet, another tooth shall begin its action on the other pallet. 

A simple form of escapement is shown in Fig. 219. The frame cc r 
c^ is arranged to slide longitudinally in 

the bearings CC, which are attached 
to the bearing for the toothed wheel. 
The wheel a turns continually in the 
direction of the arrow, and is provided 
with three teeth, b, b' ', b" ', the frame 
having two pallets, c and c'. In the 
position shown, the tooth b is just 
ceasing to drive the pallet c to the right, and is escaping, while the tooth 
b f is just coming in contact with the pallet c', when it will drive the 
frame to the left. 

While escapements are generally used to convert circular into recip- 
rocating motion, as in the above example, the wheel being the driver, 
yet, in many cases, the action may be reversed. In Fig. 219, if we con- 
sider the frame to have a reciprocating motion and use it as the driver, 
the wheel will be made to turn in the opposite direction to that in 
which it would itself turn to produce reciprocating motion in the frame. 
It will be noticed also that there is a short interval at the beginning 
of each stroke of the frame in which no motion will be given to the wheel. 
It is clear that the wheel a must have 1, 3, 5, or some odd number of 
teeth upon its circumference. 

The Crown-wheel Escapement. — The crown-wheel escapement (Fig. 
220) is used for causing a vibration in one axis by means of a rotation 
of another. The latter carries a crown wheel A, consisting of a circular 
band with an odd number of large teeth, like those of a splitting-saw, 
cut on its upper edge. The vibrating axis, 0, or verge as it is often 
called, is located just above the teeth of the crown wheel, in a plane 





ESCAPEMENTS. 147 

at right angles to the vertical wheel axis. The verge carries two 

pallets, b and b v located in planes passing through 

its axis, the distance between them being arranged 

so that they may engage alternately with teeth on 

opposite sides of the wheel. If the crown wheel 

be made to revolve under the action of a spring 

or weight, the alternate action of the teeth on 

the pallets will cause a reciprocating motion in 

the verge. The rapidity of this vibration depends 

upon the inertia of the verge, which may be ad- p IQ 2 2o 

justed by attaching to it a suitably weighted arm. 

This escapement, having the disadvantage of causing a recoil in the 
wheel as the vibrating arm cannot be suddenly stopped, is not used in 
timepieces, and but rarely in other places. It is of interest, however, 
as being the first contrivance used in a clock for measuring time. 

144. The Anchor Escapement. — The anchor escapement as applied 
in clocks is shown in the upper portion of Fig. 221. The escape- wheel 
A t turns in the direction of the arrow and is supplied with long pointed 
teeth. The pallets are connected to the vibrating axis or verge C x by 
means of the arms d 1 C 1 and efi 1} the axis of the verge and wheel being 
parallel to each other. The verge is supplied at its back end with an 
arm Cip v carrying a pin p x at its lower end. This pin works in a slot 
in the pendulum-rod, not shown. The resemblance of the two pallet 
arms combined with the upright arm to an anchor gave rise to the name 
"anchor escapement." The left-hand pallet, d 1} is so shaped that all 
the normals to its surface pass above the verge axis C v while all the 
normals to the right-hand pallet, e v pass below the axis C v Thus an 
upward movement of either pallet will allow the wheel to turn in the 
direction of the arrow, or, the wheel turning in the direction of the 
arrow, will, when the tooth b x is in contact with the pallet d lf cause a 
left-handed swing of the anchor; and when b t has passed off from d x 
and o t reaches the right-hand pallet, as shown, a right-handed swing 
will be given to the anchor. As the pendulum cannot be suddenly 
stopped after a tooth has escaped from a pallet, the tooth that strikes 
the other pallet is subject to a slight recoil before it can move in the 
proper direction, which motion begins when the pendulum commences 
its return swing. The action of the escape-wheel on the pendulum is as 
follows : 

Suppose the points l ± and k t to show extreme positions of the point p v 
and suppose the pendulum and point p A to be moving to the left ; the 
tooth b x has just escaped from the pallet d v and o t has impinged on 
e v as shown, the point p ± having reached the position m v The recoil 
now begins, the pallet e x moving back the tooth o v while pi goes from 
m x to l v The pendulum then swings to the right and the pallet e t is 



148 INTERMITTENT LINKWORK.— INTERMITTENT MOTION. 



urged upward by the tooth o lf thus urging the pendulum to the right 
while p x passes from l x to n lf when o x escapes. Recoil then occurs on 
the pallet d x from n x to k lt and from k x to m x an impulse is given to the 
pendulum to the left, when the above-described cycle will be repeated. 
As the space through which the pendulum is urged on exceeds that 
through which it is held back, the action of the escape-wheel keeps 
the pendulum vibrating. This alternate action with and against the 
pendulum prevents it from being, as it should be, the sole regulator of 
the speed of revolution of the es cape- wheel ; for its own time of vibra- 
tion, instead of depending only upon its length, will also depend upon 

the force urging the escape- 
wheel round. Therefore any 
change in the maintaining 
force will disturb the rate of 
the clock. 

145. Dead-beat Escape- 
ment. — The objectionable fea- 
ture of the anchor escapement 
is removed in Graham's dead- 
beat escapement, shown in the 
lower portion of Fig. 221. The 
improvement consists in mak- 
ing the outline of the lower 
surface, db, of the left-hand 
pallet, and the upper sur- 
face of the right-hand pallet, 
arcs of a circle about C, the 
verge axis; the oblique sur- 
faces b and / complete the 
pallets. The construction in- 
dicated by dotted lines in the 
figure insures that the oblique 
surfaces of the pallets shall 
make equal angles, in their 
normal position, with the tan- 
gents bC and fC to the wheel 
circle not shown. If we sup- 
pose the limits of the swing 
of the point p to be I and k, 
the action of the escape-wheel 
on the pendulum is as follows: 
The pendulum being in its 
right extreme position, the tooth b is bearing against the circular portion 
of the pallet d; as the pendulum swings to the left under the action of 




ESCAPEMENTS. 149 

gravity, the tooth b will begin to move along the inclined face of the 
pallet when the centre line has reached w, and will urge the pendulum 
onward to ra, where the tooth leaves the pallet, and another tooth o 
comes in contact with the circular part of the pallet e, which, with the 
exception of a slight friction between it and the point of the tooth, will 
leave the pendulum free to move onward, the wheel being locked in 
position. On the return swing of the pendulum, the inclined part of 
the pallet e urges the pendulum from ra to w. Hence there is no recoil, 
and the only action against the pendulum is the very minute friction 
between the teeth and the pallets. The impulse is here given through 
an arc raw, very nearly bisected by the middle point of the swing of 
the pendulum, which is also an advantage. The term " dead-beat" has 
been applied because the second hand, which is fitted to the escape- 
wheel, stops so completely when the tooth falls upon the circular por- 
tion of a pallet, there being no recoil or subsequent trembling such as 
occurs in other escapements. 

In watches the pendulum is replaced by a balance-wheel swinging 
backward and forward on an arbor under the action of a very light 
coiled spring, often called a " hair-spring"; the pivots of the arbor are 
very nicely made, so that they turn with very slight friction. 

146. The Graham Cylinder Escapement. — This form of escapement 
is used in the Geneva watches. Here the balance verge (Fig. 222) 
has attached to it a very thin cylindrical shell rs centred 

at 0, the axis of the verge, and the point of the tooth b can 

rest either on the outside or inside of the cylinder during 

a part of the swing of the balance. As the cylinder 

turns in the direction of the arrow (Fig. 222, A), the 

wheel also being urged in the direction of its arrow, the 

inclined surface of the tooth be comes under the edge s 

of the cylinder, and thus urges the balance onward; this 

gives one impulse, as shown in Fig. 222, B. The tooth jr IG 222. 

then passes s, flies into the cylinder, and is stopped by 

the concave surface near r. In the opposite swing of the balance the 

tooth escapes from the cylinder, the inclined surface pushing r upward, 

which gives the other impulse in the opposite direction to the first; 

the action is then repeated by the next tooth of the wheel. 

This escapement is, in its action, nearly identical to the dead-beat; 
but the impulse is here given through small equal arcs, situated at equal 
distances from the middle point of the swing. 

147. The Chronometer Escapement is shown in Fig. 223. Here the 
verge carries two circular plates, one of which carries a projection p, 
which serves to operate the detent d; the other carries a projection w, 
which swings freely by the teeth of the escape-wheel when a tooth is 




150 INTERMITTENT LINKWORK.— INTERMITTENT MOTION. 




Fig. 223. 



resting upon the pallet d, but encounters a tooth when the wheel is in 

any other position. 



The detent d has a compound 
construction and consists of four 
parts : 

1° The locking-stone d, a 
piece of ruby on which the tooth 
of the escape-wheel rests. 

2° The discharging-spring I, 
a very fine strip of hammered 
gold. 

3° A spring s on which the 
detent swings, and which attaches 
the whole to the frame of the 
chronometer. 

4° A support e, attached to 
the body of the detent, to prevent the strip I from bending upward. 
A pin r prevents the detent from approaching too near the wheel. 
The action of the escapement is as follows: On a right-hand swing 
of the balance the projection p meets the light strip I, which, bending 
from its point of attachment to the detent, offers but very little resist- 
ance to the balance. On the return swing of the balance, the projec- 
tion p meets the strip I, which can now only bend from e, and raises 
the detent d from its support r, thus allowing the tooth b to escape, 
the escape-wheel being urged in the direction of the arrow. While this 
is occurring, the tooth b 2 encounters the projection n, and gives an 
impulse to the balance; the detent meanwhile has dropped back under 
the influence of the spring s, and catches the next tooth of the wheel b v 
It will be noticed that the impulse is given to the balance imme- 
diately after it has been subject to the resistance of unlocking, the 
detent d, thus immediately compensating this resistance ; also that 
the impulse is given at every alternate swing of the balance. 

The motion of the balance is so adjusted that the impulse is given 
through equal distances on each side of the middle of its swing. 



CHAPTER X. 
WHEELS IN TRAINS. 



148. Since rolling cylinders cannot be relied upon to transmit much 
force, and at the same time maintain a constant velocity ratio, they are 
provided with teeth upon their rims, as shown in Fig. 224, which repre- 
sents a short section of the rim of a toothed wheel. When rolling 
cylinders, either external or internal, are thus supplied with teeth, they 
are called gear-wheels, or, better, spur gears) rolling cones similarly 
supplied with teeth are called bevel-wheels or bevel gears. In the latter 
case, when the axes are at right angles with each other, the bevel gears 
are commonly known as mitre gears. A small spur-wheel is called a 
opinion. 

Toothed wheels are said to be in gear when they are capable of 
moving each other, and out of gear when they are so shifted in position 
that the teeth cease to act. 

The teeth of gear-wheels are so shaped that they give, by their slid- 
ing contact, the same velocity ratio as the rolling cylinders or cones, 
from which they are derived, give by their 
rolling contact. The rolling cylinders or cones, 
as the case may be, that give the same velocity 
ratio by their rolling contact as the teeth give 
by their sliding contact, are called pitch cylin- 
ders and pitch cones respectively. The inter- 
section of the pitch cylinder with a plane per- 
pendicular to its axis is called the pitch circle. 
In bevel gearing the largest intersection of the 
perpendicular plane, or the base of the rolling 
cone, is commonly called the pitch circle. Fig. 
224 shows a pair of spur gears, the circles in con- 
tact at c being the pitch circles. The pitch of 
a gear-wheel is the distance measured on the 
pitch circle from a point on one tooth to a 
similar point on the next tooth, as ab, Fig. 224. 
In all cases the pitch must be an aliquot part of 
the pitch circle, and only wheels having the 
same pitch will gear with each other. In what 
follows, gear-wheels will be represented by their pitch circles, their pitch 
cylinders, or their pitch cones, as may be convenient in each case. 

151 




Fig. 224. 



52 



WHEELS IN TRAINS. 



149. In the preceding chapters, the elementary combinations dis- 
cussed have, in general, consisted of two principal parts only, a driver 
and a follower; and it has been shown how to connect them to produce 
the required velocity ratio. 

There may occur cases, however, in which, although theoretically pos- 
sible, it may be practically inconvenient to effect the required commu- 
nication of motion by a single combination. In such a case a series 
or train of combinations is employed, in which the follower of the first 
combination of the train is carried by the same axis or sliding piece 
to which the driver of the second is attached; the follower of the second 
is similarly connected to the driver of the third, and so on. In this 
section revolving pieces only, or trains of wheels, will be discussed. 

Fig. 225 shows an example of a train of wheels where the follower B 
and driver C are placed on the same axis, A being the first driver. The 
second follower, D, and the third driver, E, are also on the same axis. 
The numbers preceding the letters t indicate the numbers of teeth on 
the wheels upon which they are placed. 

150. Value of a Train of Wheels. — By the value of a train we mean 
the ratio of the angular velocities of the first and last axes, or, what is 
equivalent, the ratio of their rotations in a given time. Thus in the 
train shown in Fig, 225, if we let n x represent the turns of A and n 4 
the turns of F, the value of the train will be 

turns of F _n 4 
turns of A n t ' 
If we let n 2 and n 3 represent the turns of the second and third axes re- 
spectively, the value of the train could be written 

turns of F jn 2y n^ y n A _n^ - 

turns of A n x n 2 n 3 n t 

That is, the value of the train may be found by multiplying together 
the separate ratios of the synchrorial rotations of the successive pairs of 
axes. 

In § 148 it was stated that two wheels that will work together must 

have the same pitch. There- 
fore the numbers of teeth 
on any two wheels which 
will work together are pro- 
portional to the diameters 
of the respective pitch cir- 
cles. It has already been 
shown that the diameters of 
two rolling cylinders are 
inversely proportional to the 




Fig. 225. 



rotations. From this it follows that the rotations of a pair of wheels 



VALUE OF A TRAIN. 153 

are inversely proportional to the numbers of teeth on the wheels. Thus 

in Fig. 225 we should have 

t urns of £ _ 200 turns of D _ 120 turns of F ^ 80 
turns of A ~ 80 ' turns of C ~ 60 ' turns of E " 30' 

or 

turns of F 200 120 80 = 40 

turns of A ~ 80 X 60 30~ 3 * 

When the first and last axes of a train turn in the same direction the 

value of the train is considered positive, and when in opposite directions 

negative. The value of the train in Fig. 225 should then be written 

turns of F_ 

turns of A ¥ " 

[When discussing the more complex epicyclic trains in Chapter XI it 

becomes necessary to use the plus or minus signs in the solutions, as 

the algebraic sum of two or more component motions will be required.] 

Similarly, the value of the belt train shown in Fig. 226, where the 

diameters of the pulleys are given in inches, is 

turns of F 




Fig. 226. 

151. General Example. — It is not necessary that all the separate 
velocity ratios be expressed in the same terms as previously explained. 
For example, let us take a train of six axes, and let : 

1° The first axis turn once per minute, and the second once in fifteen 
seconds. 

2° The second axis turn three times while the third turns five times. 

3° The third axis carry a wheel of sixty teeth, driving a wheel of 
twenty-four teeth on the fourth axis. 

4° The fourth axis carry a pulley of twenty-four inches diameter, driv- 
ing by means of a belt a pulley twelve inches in diameter on the fifth axis. 

5° The fifth axis have an angular velocity two-thirds of that of the 
sixth axis. 

Then 

_60 5 60 24 S_ 
e ~"l5 X 3 X 24 X 12 X 2~ 50 ' 
or the last axis will make fifty turns while the first turns once. 

152. Directional Relation in Trains. — This depends on the number 
and manner of connection of the different axes. When the train con- 



154 



WHEELS IN TRAINS. 



Wf* 



Fig. 227. 



sists solely of spur-wheels or pinions on fixed parallel axes, the direction 

of rotation of the successive axes will be 
alternately in opposite directions. In any 
arrangement the directional relation can be 
traced by placing arrows on the different 
wheels, showing their direction of rotation 
and following through the entire combina- 
tion. It is frequently the case that two 
separate wheels of a train, as A and B, 
Fig. 227, may revolve about the same axis, 
and if they revolve in the same direction, 
they may be connected by means of two 
other spur-wheels revolving on an axis 
parallel to that of the wheels ; while if they 
revolve in opposite directions, Fig. 228, 
they might be connected by one bevel- wheel 
placed on an axis making an angle with 
that of the wheels. 

When an annular wheel, that is, a wheel 

having teeth on the inside of an annular ring, is used in connection with 




Fig. 228. 



N 



A 2M 

i_ 



a pinion, it will be noticed that they both 
revolve in the same direction. 

Idle Wheel. — A wheel called an idle 
wheel, which acts both as a driver and a 
follower, is often placed between two 
others. In such a case the velocity ratio 
of the two extreme wheels is not affected, 
but the directional relation is changed, the 
extreme wheels now rotating in the same 
direction, while if they geared directly, 
they would rotate in opposite directions. 

153. Examples of Wheels in Trains. — 
Fig. 229 shows the train in a cotton carding- 
machine. For the train we have the value 

tH5Lf = ^x§xMxg= + 37.84. 

turns A \( 20 26 33 

In the machine the lap of cotton passing under the roll A is much drawn 
out in its passage through the machine, and it becomes necessary to solve 
for the ratio of the surface speeds of the rolls B and A . For this we have, 
since the surface speed equals 2tt X turns X radius or turns XttX diameter, 

surface speed B _ turns ffxdiam. B 

surface speed A 



B,i" 



B 4" 



Fig. 229. 



turns A X diam. A 
surface speed B l 



(53) 



surface speed A 



37.84 



2.25 



67.27. 



EXAMPLES OF WHEELS IN TRAINS. 



155 



25"t 



A train of spur gears is often used in machines for hoisting where the 
problem would be to find the ratio of the weight lifted to the force applied. 
In Fig. 230 the value of the train is fi 

turns B 21 25 = J^. tP p 

turns A ~ 100 X 84 ~ 16 ; ? 

then, if £> = 15" and R = 2± it., F 

Lv.JP 1 ~16 X 60~64 ; 

F l.v. TF = 1 
•*' F~l.v. F ~64' 
or if f were 50 lbs., W would be 3200 lbs. if loss 
due to friction were neglected. 

Fig. 231 shows a train of wheels used to connect a set of three lower 
drawing rolls, such as used in cotton machinery for drawing cotton. 



w 
Fig. 230. 




Fig. 231. 
The figure at the left is an elevation, and the other figures are end views 
of the trains connecting the rolls C and A, and A and E, respectively. 
The top rolls placed over A, E, and C, and turning with them, are not 
shown in the figures. The cotton to be drawn passes between the top 
and bottom rolls, pressure being applied to the top rolls to keep them 
in contact with the lower ones. 

The train connecting the rolls C and A consists of four wheels, 1, 2, 3, 
and 4; that connecting the rolls A and E of three wheels, 5, 6, and 7, and 
it may easily be seen that the rolls all turn in the same direction. Sup- 
pose the roll C to, be 1J" in diameter, and A and E both 1" diameter, the 
wheel 1 to have 36 teeth; 2, 80 teeth; 3, 35 teeth; 4, 56 teeth; 5, 22 
teeth; 6, 56 teeth; and 7, 20 teeth. The value of the train will be 



turns C 
turns A 



56 80 = 32 
35 36 ~ 9' 



and 



surface speed C_32 9 
surface speed A 9 8 



4 
1' 



or the surface of C moves four times as fast as that of A. Thus a cotton 
sliver passing in between the rolls A is drawn to four times its length on 
emerging from the rolls C. This is termed the draft of the rolls. 



156 WHEELS IN TRAINS. 

The value of the train connecting the rolls A and E is 
turns E 22x56 22 



turns A 56X20 20 



1.1. 



or E turns 1.1 times while A turns once, the draft here being 1.1, since 
the rolls are of the same diameter. 

In order to arrange the rolls for different staples of cotton and dif- 
ferent drafts we must be able to change the distance between the 
axes of the rolls to suit the former, and the value of the train to suit the 
latter. The front roll C revolves in fixed bearings; the bearings of 
the remaining rolls are made adjustable, which necessitates making the 
axes of the wheels 2 and 3, and 6 adjustable. 

The value of the train connecting A and E is never changed, and it is 
only necessary here to provide a slot for the adjustable stud M, which 
furnishes the bearing for the wheel 6. The value of the train connect- 
ing C and A, however, is changed to give the required draft, and this 
is usually accomplished by arranging so that one of the wheels, as 3, 
called a change gear, can be removed easily, and replaced by another. 
The arrangement is as follows: 

An arm B, centred on the shaft C, carries a stud N on which turn the 
two wheels 2 and 3. The change gear 3 fits over the extended hub of 
the wheel 2, and is made to turn with it by means of a pin in the hub, 
a nut and washer on the end of the stud serving to keep the wheels in 
place. The arm B is provided with a circular slot whose centre is in 
the axis of C, this slot being supplied with a set-screw S in the standard 
supplying the bearing of C. After once adjusting the stud N in the 
arm B so that the wheels 2 and 1 are in proper relative position, and 
placing the necessary change wheel upon the hub of 2, the set-screw S 
being loose, the arm and wheels can be turned on the shaft C until 3 and 
4 come into gear ; then tightening the screw S will secure the whole, in 
position. The slot is made long enough to allow for the largest and 
smallest change gear 3. This principle of first adjusting a stud, carry- 
ing one or two gears, on an arm turning about one of the shafts which 
it is desired to connect, and then swinging this arm until the second 
contact is made, is often made use of in placing change gears on lathes 
and other machinery. 

The standard D furnishes the bearings for the rolls C and A, and the 
support for the arm B, and the standard F supplies bearings for all three 
rolls and a support for the adjustable stud M of the wheel 6. 

In a machine of this kind a so called draft factor is often determined, 
assuming the change gear to have one tooth only. The draft factor 
when divided by the draft desired will give the number of teeth in the 
change gear; and conversely, the draft factor divided by the number 
of teeth in the change gear will give the draft. In this case (Fig. 231) 



ENGINE-LATHE TRAIN. 



157 



the draft factor is 



56 
1 



36 X 8 -14U. 



With a draft gear of 35 teeth the draft is 

35 ' 
that is, 

surface speed C 



4. 



surface speed A 

Engine-lathe Train. — Fig. 232 gives an elevation and end view Of 
the headstock of an engine lathe, showing the method of connecting 




Fig. 232. 
the spindle or mandrel M with the lead screw L. The " back gears 7 ' G, H 
on the shaft NN have been drawn in position above the mandrel, 
instead of back of it where they are usually placed, so that one figure 
will show the whole arrangement, a convention often adopted in draw- 
ings of headstocks. 

The step pulley C turns loose on the mandrel M, and carries the gear 
F at its left-hand end; the gear wheels / and A are attached to the 
mandrel, and turn with it. The gear-wheels G and H, connected by a 
hollow shaft, turn upon an arbor whose axis is parallel to the axis of the 
mandrel. This arbor is provided with two eccentric bearings NN, so 
that by turning it slightly the gears G and H with their hollow shaft, 
commonly known as back gears, can be thrown in or out of gear with F 
and / respectively. An adjustable catch is arranged between the step 
pulley C and the wheel I, so that the wheel can be connected directly 
'o the pulley C by throwing in the catch, the back gears being out of 
gear, or indirectly through the back gears, which are now thrown in, 
the catch being adjusted so that the pulley turns freely on the mandrel. 
This catch consists of an adjustable pin, moving in a radial slot in the 
gear wheel /, it being held in position by a spring or thumb-nut? When 



158 WHEELS IN TRAINS. 

the catch is at the inner end of the slot, it works in a circular groove in 
the pulley C; but when it is at the outer end of the slot, it is located 
in a radial notch cut from the groove, thus compelling C and I to move 
together. 

Thus, by combining a four-step pulley with back gears, a series of 
eight speeds can be obtained for the mandrel — four with the back 
gears in, and four with the cone only. The cone and back-gear train 
should be so proportioned that the speeds are in geometric progression. 

If it is desired to have a certain value for the train from F to I, it 
becomes necessary to find suitable numbers of teeth for the four gears 
F, G, H, and I. It is to be noticed that, since the axes M and N are 
parallel, the diameter of F + diam. of G will equal diam. of H + diam. 
of I; therefore, if the same pitch were to be on all four gears, we should 
have teeth on F + teeth on(?= teeth on H + teeth on I. In a lathe the 
pitch on H and I would usually be a little greater than on F and G, since 
the teeth on H and 7 bear a greater stress. 

For example, let the value of the train be 

turns I _ 1 
turns>~10 ; 
this ratio may be separated into two ratios nearly alike; thus, 

turns I _ 1 _ 3 1 . 

turns F 10 10 A 3 ; K J 

and if we are to use not less than 24 teeth on any wheel, the train could 
be as follows: 

turns I teeth on F teeth on H _ 3 1 24 24 
turns F "teeth on G teeth on Z"10 3 80 72' 

which would be a train suitable for back gears. If it were required 
that the pitch should be the same on all the gears, the train could be 

s4-S x S where 24 + 80 = 26+78 - _ 

This same type of train occurs in 
clockwork between the minute and hour 
hands. The value of the train (Fig. 233) 
would be 

tums_M = 12_3 4^30 32 
turns H 111 10 X 8' 
or, letting A, B, C, and D represent 
the numbers of teeth, 

A C 10 8 1 turns H 



D 
C~~ 



Fig. 233. 
In this arrangement the wheels are all of the same pitch. 



B X D 30 32 12 turns AT* 



SCREW-CUTTING. 159 

154. Screw-cutting. — One of the most frequent uses of an engine 
lathe is that of cutting screws. To this end, the lathe is provided with 
an accurately cut screw L (Fig. 232) , called a lead screiv, which serves to 
move the carriage of the lathe by turning in a nut P made fast to the 
carriage. This carriage travels upon the bed of the lathe, on ways 
parallel to the axis of the mandrel, and carries the tool-holder; the 
whole is here represented by the pointer R, which illustrates the action 
just as well. 

A train of gear-wheels connects the mandrel with the lead screw, an 
intermediate shaft, as that carrying the wheels B and C, commonly 
known as the "stud," being generally used. This stud is then connected 
to the lead screw by means of the train CDE, D being an idle wheel, 
and C and E change gears. Let us suppose that the pitch of the lead 
screw L is 1" L.H. and that we wish to cut a screw S having ten threads 
per inch R.H. While the lead screw turns six times the cutting-tool 
represented by R will travel one inch ; therefore the screw $ must make 
ten turns in the same time. 

turns S _ 10 _ teeth on E teeth onS ( 

turns L 6 teeth on C teeth oni' * ^ ' 

or, we may always write 

turns S X pitch of S = turns L X pitch of L, ... (56) 
from which 

1 
turns S _ 6 10 
turns L 1 6 ' 
10 

The numbers of teeth on A and B between the mandrel and the stud 
would be known, and it would be necessary to find suitable numbers of 
teeth for the gears C and E, and also to determine whether an odd or 
an even number of idlers D would be needed to give the desired thread. 
If A and B have respectively 24 and 36 teeth, 

turns S _ teeth on E 36_10. 
turns L ~ teeth on C X 24 _ ~6 ' 
teeth on E _ 10 40 
* ' teethonC _ ¥~36 ; 

or the wheel on the stud could have 36 teeth and that on the lead screw 
40 teeth. In this way a table may be calculated for any lathe and 
the gears for each pitch of screw to be cut, arranged in tabular form, 
as given below for the above case. 



160 WHEELS IN TRAINS. 



Threads per Inch 


Gear on Stud, 


Gear on Lead ScreWj 


to be 


cut. 


C. 


E. 


3 




72 


24 


4 




72 


32 


5 




72 


40 


6 




36 


24 


7 




36 


28 


8 




36 


32 


9 




36 


36 


10 




36 


40 


etc. 




etc. 


etc. 



In arranging such a table the same gears are used as often as possible, 
and so planned that both gears need not be changed any oftener than 
is necessary. 

It will be noticed that when the screws S and L revolve in the same 
direction, the threads will both be either right- or left-handed; while if 
they revolve in opposite directions, as in the figure, one screw must 
be right- and the other left-handed. In the figure, a right-handed screw 
is being cut, the lead screw in this case being left-handed. To cut a left- 
handed screw, another idle wheel should be inserted in the connecting 
train. In many lathes it is arranged that either one or two idle wheels 
can be thrown between A and B at pleasure, by simply moving an arm 
placed near B in the headstock casting. 

In gearing the stud with the lead screw, a vibrating slotted arm, 
similar to that described in connection with the roll train, is made use 
of. The wheel C is first placed on the stud, and then E is placed on 
the lead screw; a wheel D is then selected from among the change 
gears, and placed on the movable stud of the arm W, the stud being 
adjusted so that C and D gear with each other; the arm is then swung 
over until D and E gear with each other, and clamped in position by 
means of the screw T. 

155. Clockwork. — A familiar example of the employment of wheels 
in trains is seen :'n clockwork. Fig. 234 represents the trains of a com- 
mon clock; the numbers near the different wheels denote the number 
of teeth on the wheels near which they are placed. 

The verge or anchor vibrates with the pendulum P, and if 

we suppose the pendulum to vibrate once per second, it will let one 

tooth of the escape- wheel pass for every double vibration, or every 

two seconds (§ 144). Thus the shaft A will revolve once per minute, 

and is suited to carry the second hand S. 

turns \ 

The value of the train between the axes A and C is t = 

turns A 

8x8 1 
"fkztw-a = ™, or the shaft C revolves once for sixty revolutions of A; it 
60X64 60 



CLOCKWORK. 



161 



st 



60 t 



8t 



64 t 



2S 



64 t 

r 

t 

l" 

8t 



is therefore suited to carry the minute hand M. The hour hand H is also 
placed on this shaft C, but is at- 
tached to the loose wheel F by 
means of a hollow hub. This wheel 
is connected to the shaft C by 
means of a train and intermediate 
shaft E. The value of this train is 

turns F _ 28 X 8 _ 1 
turns M ~~ 42 X 64 ~ 12/ 

The drum D, on which the 
weight-cord is wound, makes one 
revolution for every twelve of the 
minute hand M, and thus revolves 
twice each day. Then, if the 
clock is to run eight days, the 
drum must be large enough for 
sixteen coils of the cord. The 
drum is connected to the wheel 
G by means of a ratchet and 
click, so that the cord can be 
wound upon the drum without 
turning the wheel. 

Clock trains are usually ar- 
ranged as shown in the figure, 
the wheels being placed on shafts, 
often called " arbors," whose 
bearings are arranged in two 
parallel plates which are kept 
the proper distance apart by 
shouldered pillars (not shown) 
placed at the corners of the plates. 
When the arbor E is placed 
outside, as shown, a separate 
bearing is provided for its outer 
end. 

156. Frequency of Contact be- 
tween Teeth. Hunting Cog. — Let 

T and t be the numbers of the 
teeth on two wheels in gear, and 

'let — = - when reduced to its low- 
t 



42 t 



96 t 



Fig. 234. 
est terms. It is evident that the 

same teeth will be in contact after a turns of t, and b turns of T. 

Therefore the smaller the numbers a and b, which express the velocity 



162 WHEELS IN TRAINS. 

ratio of the two axes, the more frequently will the same pair of teeth 
be in contact. 

Assume the velocity ratio of two axes to be nearly as 5 to 2. Now 
if we make T = S0 and t = 32, we shall have 

7 = 32 = 2' exaCtly ' 

and the same pair of teeth will be in contact after five turns of t or two 

turns of T. 

T 81 5 
If we now change T to 81, then 7=^5 = ft, very nearly, the 

angular velocity ratio being scarcely distinguishable from what it was 
originally. But now the same teeth will be in contact only after 81 
turns of t or 32 turns of T. 

The insertion of a tooth in this manner prevents contact between 
the same pair of teeth too often, and insures greater regularity in the 
wear of the wheels. The tooth inserted was called a hunting cog, because 
a pair of teeth, after being once in contact, would gradually separate 
and then approach by one tooth in each turn, and thus appear to hunt 
each other as they went round. In cast gears, which will be more 
or less imperfect, it would be much better if an imperfection on any 
tooth could distribute its effect over many teeth rather than that all 
the wear due to such imperfection should come always upon the same 
tooth. This result is most completely obtained when the numbers of 
teeth on the two gears are prime to each other, as above when T and 
t were 81 and 32 respectively. 

This same principle is used in a form of stop motion. Suppose 
the wheel A, Fig. 235, to have 61 teeth and B to have 30 teeth. The 

same teeth will be in 
contact after 30 turns of 
A or 61 of B. If now 
one of the wheels A is 
arranged on a movable 
axis and two false teeth 
are supplied on the 
wheels so as to meet at 
a point, one wheel can 
be made to push the 
other one side and oper- 
ate a stop motion for a 
machine after a certain desired motion, as 61 turns in the above ex- 
ample, of the fixed wheel B has taken place. 

Where it is necessary to have some exact value to the train, as in 
clockwork, the above principle can only be employed to a limited 
extent. 




Fig. 235. 



METHODS OF DESIGNING TRAINS. 163 

157. Methods of Designing Trains. — Let it be required to connect 
two axes, A and B, by a train of spur gears so that the value of the 
train shall be 

turns B_sy 

turns A ' 

and further let the largest number of teeth allowed on any wheel be T 

and the smallest number be t. Then one pair of wheels using teeth T 

T 
and t respectively would give an a.v. ratio — to the axes thus connected. 

It is first necessary to determine the least number of pairs of gears 

required to obtain the value of the train. This is easily done by finding 

T 
the power to which the ratio — must be raised to give a result not less 

z 

than the value of the train. If m = the number of pairs of gears, 



=© m ; 



« = ^J (57) 

log T 



_ , turns B ,__ , T 150 , ,. /r _ x 

For example, suppose -r = 400, and -r=-r^~; from equation (57), 

turns JL t Zd 

m =!2!*» 3.344; 

log 6 

or, three pairs is not sufficient, therefore four pairs of gears are required. 

T . 
Where the ratio — is some number easily raised to successive powers 

the least number of pairs can be obtained without using equation (57). 

T 

Thus, in the above example, 7 = 6; 6X6X6 = 6 3 = 216, the result of 

z 

using three pairs, which is not enough; 6 4 = 1296 is much more than 
enough, but four pairs are required, and the fact that four pairs would 
give so much more than is desired means simply that we can use less 
than 150 teeth on the large wheels. 

After determining the least number of pairs of gears needed, one of 
two methods may be used in solving for the desired train, the choice 
depending largely on whether it is required that the desired value of 
the train shall be exactly obtained, or whether some deviation is allow- 
able. 

T 120 
1° Let the value of the train be 360, and - = ~^7T = 6. The least 

number of pairs of gears required will be four (since 6 3 = 216<360). 



164 WHEELS IN TRAINS. 

Separate 360 into its prime factors, and place these factors in the numer- 
ator, letting the denominator be written as shown, the four units repre- 
senting the four small gears in the final train; thus, 

360 = 2-2-2-3-3-5 
1 " lll-l ' 

Since 20 is the least number of teeth to be used, we may next substi- 
tute for the four units four twenties, multiplying the numerator also by 
the same amount, giving 

360 ^ 2-2 -2 -3 -3 -5-20 -20-20-20 
i . 20-20-20-20 

The factors of the numerator are now to be combined into four numbers 
which are to be the teeth for the large gears, and to obtain the best 
result the twenties in the numera or should be factored as well as the 
360. Two results are given below: 



or, better, 



360 = 100 100 90 64 
1 20 20 20 20' 



360^100 96 80 75 
1 ~20 X 20 X 20 X 20' 



If, however, it be required that the numbers of teeth on the wheels 
which are in gear should be prime to each other, it will be necessary 
to choose such numbers of teeth for the small gears as will give when 
combined with the factors of the value of the train numbers which can 
fulfil the requirements. 

rn -j rr\ 

For example, let the value of the train be 400 and — = -^ =6. Four 

pairs of wheels will be found necessary. Then we may write 

400 2 2 -2 -2 -5- 5 -27-27 -26-26 
1 " 27-27-.26-26 

_104 104 135 135 

~ 27 X 27 X 26 26 * 

For the other train, when the value was 360, with prime gears, and to 
have the large gears as small as possible, the small gears will probably 
need to be all different; thus, 

360 _ 2-2-2-3-3-5-20- 21 -' (2- 11) -23 
1 " 20-21-22- 23 

= 99 92 105 80 
20 21 22 23' 



METHODS OF DESIGNING TRAINS. 165 

Should the numerator or denominator of the fraction expressing the 
value of a train be a large prime number, or contain inconveniently 
large prime factors, then an approximate ratio can be sought having 
terms that can be factored. 

In an orrery the earth must rotate on its axis once in 24 hours = 
86400 seconds, and revolve round the sun in 365 days 5 hours 48 min- 
utes 48 seconds = 31556928 seconds. Then the value of the train will be 

31556928 = 164359 = 269X4 7X13 

86400 ~ 450 10X9X5 

Here 269 is a large prime number, and to avoid its use let us intro- 
duce in its place 

o«nnm 269001 81X81X41 .„ . 

269 - 001 = ^oo(T = ioxioxi(T 269 ' very nearly ' 

Then arranging the factors, we obtain the following train: 
81 81 41 47 13 nntrnAnn 

io x ro x ro x 45 x ro= 365 - 2436 ' 

which gives a very close approximation. 

This principle of adding a very small amount to one of the prime 
factors of the value of a train, in order to obtain a number which can 
be factored, may be applied in many cases • the conditions in each case 
can alone determine whether the variation is permissible or not. 

2° Where an error of a certain amount is allowable, as would very 

often be the case, the following method may be used to advantage. 

T 100 
For example, let the value of the train be 60 and — = --- = 5. It 

t Zi 

will be found that three pairs of gears are needed. Therefore take the 
cvibe root of 60, which is 3.91+ , and write 

3.91 3.91 3.91 nn 

— X^j-X— j— = 60, nearly. 

Since the small gears are not to have less than 20 teeth, and since 
20X3.91 = 78+, we may write as a first approximation 

79 79 79 
20 20 X 20' 

which will be found to equal 61.63; if this result is too greatly in error, 
a reduction of one or two teeth in the numerator or an increase in the 
denominator may give a closer result, as 

77 79 79 
20 X 20 X 20 = 60 - 07 - 



166 



WHEELS IN TRAINS. 



158. Mangle- wheels. — Mangle- wheels are used to produce reciprocat- 
ing motion from the uniform rotation of a pinion, and derive their 
name from the first machine in which they were applied. 

Figs. 236, 237, and 238 show three forms of mangle-wheels. In 
Fig. 236 the teeth are drawn in on only a part of the pitch curve PP r 
and in Fig. 237 the pitch curves only are shown. 




Fig. 236. 



Fig. 237. 



In Figs. 236 and 237 the cycle of motion of the wheel is divided into 
two parts, each part having its own definite velocity ratio, which is here 
constant except for a small space at each end of the motion, when the 
pinion is being guided from one pitch circle P to the other, they being 
joined at their ends by short circular arcs. 

Fig. 236 has the teeth cut upon the edge of an annular groove in the 
disc A, these teeth being properly formed to gear with the pinion P, vhe 
shaft of which is so supported as to allow the pinion to gear with both 
the inner and outer sides of the groove. The pinion's shaft projects 
below the pinion, and works in a groove, the width of the groove being 
a little greater than the diameter of the pinion's shaft. This groove 
serves to keep the pinion always in gear. If we suppose the pinion to 
rotate right-handed, the wheel A will first make about f of a rotation 
left-handed, and then about one rotation right-handed, and so on. 
It will be noticed that the change of motion is gradual at each end when 
the pinion is passing from one position to the other. 

In Fig. 237 the wheel A (only one-half of which is shown) has teeth 
cut upon the outside of an annular ring projecting from the face of A, 
the pinion now travelling outside of the pitch circle PP, & groove being 
supplied for the shaft as before. Here the difference between the veloc- 
ity ratios is less than in the previous case, and by making the two 



MANGLE-WHEELS. 



167 



pitch lines PP to coincide, as has been done in Fig. 238, the velocity 
ratios are made the same. 

In Fig. 238 the wheel A is supplied with a series of pins P, the pin- 
ion B working alternately on the inside and then on the outside of the 
pins. One method of connecting 
the pinion to a fixed shaft I is 
shown (see also Fig. 236). The 
arm D, turning or the shaft 7, 
carries the pinion shaft H and a 
train of gear-wheels EFG, con- 
necting the shafts H and I ; these 
gear-wheels, being above the 
arm, are shown dotted. The 
shaft H projects below the pinion, 
and can only move between the 
positions H and H l ; the guards J J 
against which it comes when the 
last pin is reached serve to guide 
it from one side of the teeth to 
the other. As the teeth of the 
pinion cannot be shaped to give 

an exact velocity ratio, and at the same time work both on the inside 
and outside of the pins, this arrangement cannot be used where great 
uniformity of motion is desired. 

The wheels illustrated have their teeth arranged in circular rows, 
but this is not necessary, as by properly curving the rows of teeth the 
velocity ratio may be varied at pleasure; a pause can also be introduced 
by making the row radial for the corresponding distance. 

As examples of the use of mangle- wheels, we have the "flat strip- 
ping device" on flat carding-engines, and the mechanism for moving the 




Fig. 238. 




Fig. 239. 



Fig. 240. 



rails up and down in "spoolers," used for winding warp yarn from 
the bobbins on to the spools. 

Figs. 239 and 240 show two forms of mangle-racks, which are let- 
tered similarly to the wheels, the shafts of the pinions being constrained 
to move in vertical lines. 

In Fig. 240 the pinion B will gear correctly with the circular pins P, 
and in both cases the velocity ratios will be the same. 



V6S WHEELS IN TRAINS. 

Sometimes the pinion is fixed and the rack shifts laterally, it being 
arranged to move in suitable guides or to be governed by linkages 
properly arranged. 

Mangle-racks are used in some forms of cylinder printing-presses to 
actuate the table. 



CHAPTER XI. 



AGGREGATE COMBINATIONS. 



159. Aggregate Combinations is a term applied to such assemblages 
of pieces in mechanism in which the motion of the follower is the result- 
ant of the motions given to it by more than one driver. The number of 
independently-acting drivers which give motion to the follower is gener- 
ally two, and cannot be greater than three, as each driver determines 
the motion of at least one point of the follower, and the motion of three 
points in a body fixes its motion. 

By means of aggregate combinations we may produce very rapid or 
slow movements and complex paths, which could not well be obtained 
from a single driver. 

-Figs. 241 and 242 repre- 



Qa 






•fOc 



na 






bb 



\\ 



\ \ / 

\ V 
\ I 

61^ — tyb 

\ 1 
\ 1 

\ 1 
\ l 
\ 



**, 



\i 



160. Aggregate Motion by Linkwork. 

sent the usual arrangement of 
such a combination. A rigid bar 
ab has two points, as a and b, 
each connected with one driver, 

while c may be connected with a^ 

a follower. Let aa A represent the \\ 
l.v. of a, and bb x the l.v. of b: \ x 

to find the l.v. of c. Consider the c h 

motions to take place separately; 
then if b were fixed, the l.v. aa ± 
given to a would cause c to have 
a velocity represented by cc v 
Considering a as fixed, the l.v. 
bb t at b would give to c a velocity 
cc 2 . The aggregate of these two would be the algebraic sum of cc t and 
cc 2 . In Fig. 241 we have cc x acting to the left, while cc 2 acts to the 
right; therefore the resulting l.v. of c will be cc^ = cc t — cc 2 acting to the 
left, since cc x >cc 2 . In Fig. 242, where both cc x and cc 2 act to the left, 
the result is cc z = cc^-\-cc 2 acting to the left. It will be seen that the 
same results could have been obtained by finding the instantaneous 
centre of ab in each case, when we should have l.v. c:l.v. a = co:ao. 
In many cases the lines of motion are not exactly perpendicular 
to the link, nor parallel to each other, neither do the points a, b, and c 
necessarily lie in the same straight line, but often the conditions are 
approximately as assumed in Figs. 241 and 242, so that the error intro- 

169 



-o 



Fig. 241. 



Fig. 242. 



170 



AGGREGATE COMBINATIONS. 



duced by so considering them may be sufficiently small to be practi- 
cally disregarded. 

As examples of aggregate motion by linkwork we have the different 
forms of link motions as used in the valve gears of reversing steam 
engines. Here the ends of the links are driven by eccentrics, and the 
motion for the valve is taken from some intermediate point on the link 
whose distance from the ends may be varied at will, the nearer end 
having proportionally the greater influence on the resulting motion. 

A wheel rolling upon a plane is a case of aggregate motion, the centre 
of the wheel moving parallel to the plane, and the wheel itself rotat- 
ing upon its centre. The resultant of these two motions gives the 
aggregate result of rolling. 

161. Pulley-blocks for Hoisting. — The simple forms of hoisting- 
tackle, as in Fig. 243, are examples of aggregate combinations. The 




w 

Fig. 243. Fig. 244. 

sheaves C and D turn on a fixed axis, while A and B turn on a bearing 
from which the weight W is suspended. Fig. 244 is in effect the same 
as Fig. 243, but gives a clearer diagram for studying the l.v. ratio. As- 
sume that the bar ab with the sheaves A and B and the weight W has 
an upward velocity represented by v. The effect of this at the sheave A, 
since the point c at any instant is fixed, is equivalent to a wheel rolling 
on a plane, and there would be an upward l.v. at d = 2v. At the sheave 
B there is the aggregate motion due to the downward l.v. at e = 2v and 
the upward l.v. of the axis b = v, giving for the l.v. of /, 4v upwards. 

lv.F _4_W 
*'• }.v.W~l~ F' 



DIFFERENTIAL PULLEY-BLOCK. 



171 



Many elevator-hoisting mechanisms are arranged in a similar man- 
ner, the force being applied at W, and the resulting force being given at F. 
This means a large force acting through a rela- 
tively small distance, producing a relatively small 
force acting through a much greater distance. 

162. Differential Pulley-block. — Fig. 245 
shows a diagram of the Weston differential 
pulley-block, which consists of a double chain 
sheave A turning on a fixed axis c with a single 
sheave B below it. The chain is endless, passing 
around the larger circumference of A with the 
radius ac, then down and around B, whose radius 
is a mean between the two upper radii, so that 
the chain hangs parallel; from B the chain 
passes up and around the smaller circumference 
on A, the radius of which is be. To find the l.v. 
of TT T if the l.v. of F is represented by v, lay off 
the velocity v downward at the point a as shown. 
The chain leaving the smaller sheave at b will 



have a downward 1. 



represented by v t = v — . 

The effect of these velocities on the lower 
sheave will be to give an upward velocity v 
at d, and a downward velocity v x at e. The 
aggregate of these two will give for a resultant 

v v 1 
2~~2 



at the axis / a velocity v 4 

bc\ 
ac J' 



v? — v. t = - 




and since v 1 



have 



l.v. 1F-- 



l.v. W 
l.v. F 



_bc\ 
ac) 



ac—bc_ 

9 2ac 

we should have 



z 

w 

l.v. 



W 7.5-7 



(58) 

= ± 

30' 



For example, if ac = 7¥' and be = 7' , 

l.v. F 15 

and a force of 100 lbs. at F would raise a weight of 3000 lbs. at W if 

friction were neglected. 

The motion of W might have been derived as follows: Suppose the 
wheel A to turn once L.H.; then an amount of chain 2nac will be taken 
up at d, and an amount 2nbc will be delivered at e, thus giving a motion 
to W equal to n{ac — be); thus the l.v. ratio is 

l.v. W _7t{ac -~ be) _ac — bc 
l.v. F ~ 2nac ~ ~ 2ac ' 

One great advantage of this block is that it will retain the load. The 
pull at d, having but little more leverage than the equal pull at e, in the 



172 



A GGREGA TE COM BIN A TIONS. 




block as practically constructed, is not sufficient to more than overcome 
the friction of the chain and bearing c. 

163. Epicyclic Trains may be defined as trains of wheels in which 
some or all of the wheels have a motion compounded of a rotation about 
an axis and a revolution, or translation, of that axis. 

The wheels are usually connected by a rigid link, such as D (Fig. 
246). This link is usually called the train arm, and it often rotates 

upon the axis of the first wheel 
A of the train: the last wheel 
of the train may or may not be 
placed upon this axis. 

In what follows we will con- 
sider a wheel to have made one 
turn or rotation when an arrow 
placed upon it, as in Fig. 246, 
comes again into a position par- 
allel to its first position with the 
head upward, after a continu- 
ous angular mot ion, either right- 
or left-handed. [The word turn 
will be used in place of rotation, 
as it is much shorter; right-handed turns will be considered positive, and 
left-handed turns negative.] In Fig. 246 the wheel A is assumed to be 
fixed, and for +-J- of a turn of the arm D, as shown, the wheel B has 
turned a little more than -j- \, as indicated by the position of the arrow 
de, while the wheel C has not turned, the arrow be having moved par- 
allel to itself. 

164. There are two methods by which these trains may be solved: 
First. By resolving the resultant motion into its components and 

then allowing these to take place in succession, the motion of the arm 
being considered first. 

In Fig. 247 let A have 100 teeth and B 50; also let A make +5 
turns about the fixed axis a while the arm D makes —6 turns; to find 
the number of turns of B. 

1° Suppose the mechanism locked so that 
the wheels cannot turn in relation to the arm, 
and then turn the whole —6 turns about a, the 
number of turns which the arm is to make. This 
is expressed by the first line of the following 
table. 



Fig. 246. 




Fig. 247. 



2° Unlock the train, and put the wheel for which the motion is given, 
in this case A, where it should be. A is to make +5 turns, but it has 
been turned — 6 turns with the arm when the train was locked ; it must 
therefore be now turned -f 11 in order that its resultant turns shall be + 5. 



E PI CYCLIC TRAINS. 173 

This will cause B to make -22 turns, since the value of the train 

- ul ns 4 = — 2. These motions are expressed by the second line of the 
turns A 

table. 

3° Taking the algebraic sum of the above component motions will 

give the resultant motions, as expressed by the third line of the table. 

A B Arm 

1° Train locked - 6 - 6 -6 

2° Train unlocked, arm fixed +11 22 



3° Resultant turns + 5 -23 -6 

If in the same train it were required to find the number of turns which 

the arm must make in order that B shall make —28 turns, while A makes 

+ 5, we may let x be the number of turns of the arm and proceed as above. 

1° With the train locked, turn the whole x turns. 

2° With the train unlocked, and arm fixed, turn A ( — # + 5), which 

will cause B to turn ( — 2) (5 — x). 

3° Adding these component motions and equating the turns of B 
thus found with the given number of turns will give an equation from 
which x may be found. These steps are expressed in the following table: 

A B Arm 

1° Train locked x x x 

2° Train unlocked, arm fixed. . - x +5 ( - 2) (5 - x) 



3° Resultant turns +5 (—2)(5 — x)+x x 

But turns of B = -28=(-2)(5-x) +x; 

.;. x= —6 = turns of arm. 
Second Method. — By the use of a general equation which may be 
written for any of these trains, which is usually the more convenient 
method. 

In the problem just solved we could write the following general 
statement, in the form of an equation, and a similar statement could 
be made for any epicyclic train: 

turns of B relative to arm absolute turns of B — turns of arm 
turns of A relative to arm absolute turns of A — turns of arm' 
In this equation it will be seen that the. first term can always be ex- 
pressed in terms of the numbers of teeth on the gears, as it is simply 
the value of the train assuming the arm fixed. It is absolutely essential 
that this value of the train be expressed as + or — , depending on whether 
the gears considered turn in the same or in opposite directions relative to 
the arm. 

Substituting the data of the first problem, given under the tabular 
method, would give 

relative turns B _ absolute turns B — turns arm _ B — ( — 6) 

relative turns A absolute turns A — turns arm 5 — ( — 6) ' 



174 



AGGREGATE COMBINATIONS. 



where B in the last term represents the absolute turns of B; therefore 

B= -28 turns. 
With the data as given in the second case under the tabuiar method, 
where x represents the turns of the arm, and B and A the absolute 
turns of B and A respectively, 

2_ £-arm _ -28-a: t 
1 A — arm 5 — x ' 
-10+2z=-28-a;; 

x= — 6 turns. 

165. Problems of this kind may have the data so given that the 
numbers of teeth on the gears or on some of 
them may be required. Thus in Fig. 248 sup- 
pose that A is to make +5 turns, and B —28, 
while the arm turns —6; to find the number 
of teeth needed on the wheel B if A has 100 
teeth, and to determine whether or not an idle 
wheel is needed between them. By the general 




Fig. 248. 



equation, 

relative turns B 



absolute turns B — turns arm — 28 +6 



■2, 



relative turns A absolute turns A — turns arm 5 +6 
or the value of the train relative to the arm must be —2 ; so B must have 
50 teeth, and no idle wheel is required, since the value of the train is 
negative. 

166. Examples of Epicyclic Trains. — Fig. 249 shows an application 
of the two-wheel train commonly known as the Sun and Planet Wheels, 
first devised by Watt to avoid the use of a 
crank, which was patented; but in his device 
the pin b worked in a circular groove around 
the centre a, which took the place of the link ab, 
and kept the two wheels in gear. The rod B 
is attached to the beam of the engine, and a 
represents the engine shaft. While in this 
case we cannot say that the wheel C does not 
turn, yet its action on the wheel D, for an 
interval of one rotation of the arm, is the 
same as though it did not turn, as the position 
at the start and stop is the same. Then to 
find the turns of D, for one turn of the link ab 
R.H., we will first disconnect the rod and lock the train, thus obtaining 
the first line of the following table. Then, as C has been turned +1, 
we unlock the train, fix the arm, and turn C — 1, giving the second line. 




Fig. 249. 



EXAMPLES OF EPICYCLIC TRAINS 



175 



C D ab 

1° Train locked +1 +1 +1 

2° Train unlocked, arm fixed —1 ' + 1 

3° Resultant motions. . . + 2 +1 

Adding, we find that the wheel D makes 2 turns R.H. 

In the three-wheeled train, Fig. 250, let A have 55 teeth, and C 




Fig. 250. 

have 50. A does not turn; find the turns of C while the arm D makes 

+ 10 turns. By the tabular method we have: 

A C Arm. 

1° Train locked +10 +10 +10 

2° Train unlocked, arm fixed -10 —10 (f J) 

3° Resultant motions - 1 +10 

Or the wheel C turns — 1 while the arm D turns + 10. 

This type of train is made use of in rope-making machinery, to give the 
bobbins which carry the strands such a motion that the strands are not 
untwisted in laying the rope, the twist 
in laying being opposite to that of the 
strands; also in wire-rope machinery, 
where the individual wires cannot be / v_ac 
twisted in the laying. The arrangement / >a x c 

usually adopted is shown in Fig. 251. ( a # )fo) : 

The bobbins for the strands, or wires 
(shown dotted), are attached to the \ /jb"^ 
wheels B by means of the spindles b, turn- 
ing in bearings in the large disc which 
turns with the rope-laying block on the 
shaft a. The idle wheels C, turning on FlG - 251 ' 

pins placed in the disc, connect the wheels B with a stationary wheel A, 
on the axis a, having the same number of teeth as the wheel B for wire 
rope. Then, on rotating the disc, the bobbins will be carried round 
with the laying block, but the wires will not twist. This arrange- 
ment, it will be noticed, gives the same result as that shown in Fig. 115, 
and both are used for the same purpose. 

Fig. 252 shows an epicyclic train which gives the same result as 
above when the pulleys A and B, which are here connected by a belt, 




17G 



A GGREGA TE CjMBINA T10NS . 



are of the same size. This train has also been used in fibre rope-making 
machinery, the pulleys B being often made slightly smaller in diameter 



'*>- n AA 


, A 


^i 


K 


c 






^=^> 


E .... 


'! 01 


| 






r 




60 t 




60 


r 1 


"-> ^ D.' >^ 


\ XX 




"-•o;^ 


1 






1 


^P~ 


G \^^ 


D 

3. 






i 


1 




Fig. 252. 






Fig. 


25 





than A, when it will be found that they will turn slowly in the opposite 
direction to that of the arm D, giving a slight additional twist to the 
strands of the rope as they are being laid, and making a rope less liable 
to untwist. 

If the wheel C in Fig. 250 has more teeth than A, it will turn in the 
same direction as the arm. Fig. 253, called Ferguson's paradox, shows 
an arrangement giving the three cases, the wheel E having the same 
number of teeth as A, C one more than A, and F one less than A, B 
being an idle wheel connecting A with the other three. The arm D turns 
freely on the axis of the stand G, while A is fast to the stand. If D 
makes +1 turn, we have for the other wheels, by the tabular method: 

A D C E F 

1° Train locked +1 +1 +1 +1 +1 

2° Train unlocked, arm fixed. . . . -1 — g-fr -1 -gf 

3° Resultant motions +1 +^ r -^ 

Thus, C turns slowly R.H., and F 
slowly L.H., in respect to the wheel E, 
all gearing with the pinion B. 

167. The train may be a compound 
train, and also the last wheel of 
the train may turn about the same axis 
as the first, as shown in Fig. 254. Let 
the numbers of teeth be as indicated on 
the wheels ; to find the number of turns 
of A while D makes +15 turns, and 
the arm makes —3 turns; also to find 
the turns of BC. Using the tabular 
method we have: 

A BC D Arm. 

-3 -3 -3 -3 

(-«)(18) +18 





Pq 




rf=fi 




1 ! ! 




1 ! 1 




till 




Ifll 




1 D 25 1 


1 c 


69 1 !i 




|o[ 




■ 'N 1 , 


1 * 


75 t 




1 B ! ! 23 t j 




^ 




1 — ' 




1° Train locked 

2° Train unlocked, arm fixed +2 

3° Resultant motions -1 - Vs 9 + 15 ~ 3 

Or A will make —1 turn, and the sleeve on which the wheels B and 
are fast will make — 9-J-f turns. 



EXAMPLES OF EPICYCLIC TRAINS. 



177 



1 68. Annular wheels frequently enter into the construction of these 
trains, and the first and last wheels of the 
train would usually turn about the same 
axis. In Fig. 255 let A have 20 teeth and 
C have 200 teeth; C is to make —16 turns 
while A makes +50 turns. To find the num- 
ber of turns which the arm D must make. 
Using the general equation for this, 

relative turns A 



relative turns B 


absolute turns A 


— turns D 


absolute turns B 


— turns D 


10-- 


50-x 
-16-x' 


X — 


-10. 



10 



50-x 
-16-3' 




Fig. 255. 



Therefore the arm D must make — 10 turns. 

The Triplex Pulley-block, Fig. 256, is an example of the use of an 
annular epicyclic train. Here the annular D is made fast to the casing 
E so that it does not turn. The shaft H turns in bearings in the block 
frame or casing E, its axis coinciding with that of the annular D. The 
gears B and C (of which there are three equidistant sets to distribute the 




Fig. 256. 



stress evenly about the axis) connect A with D and turn loosely on 
studs fast to the disc G, which in turn is made fast to the hollow shaft of 
the chain wheel G turning in bearings as shown in the casing E. A 
hand chain wheel K is attached to the end of the shaft H, turning loosely 
in the hollow shaft of the chain wheel G, and serves to actuate the train. 



178 



AGGREGATE COMBINATIONS. 



6 


-. 


\ 


! G 

II 

! ! G 


/ 

E. 

\ 


C 




D 

/ 



Fig. 257. 



The problem is to find the l.v. ratio of W to F. This involves finding 
first the a.v. ratio of G to K. 

It will be seen that G is the arm; D, the fixed annular, is the last 
wheel and A the first wheel of the train ; the arm turns the lifting chain 
wheel, and the hand chain wheel K turns A. It is thus necessary to 
find the number of turns of G for one turn of A. Knowing the a.v. 
ratio of G to A, and the diameters L and M of the chain wheels, we may 
obtain the l.v. ratio of the chains. 

169. Epicyclic Bevel Trains. — Fig. 257 represents a common form 
of epicyclic bevel train, consisting of the two bevel-wheels D and E 

attached to sleeves free to turn 
about the shaft extending through 
> them. This shaft carries the 
cross at F which makes the 
bearings for the idlers GG con- 
necting the bevels D and E (only 
one of these idlers is necessary, 
although the two are used to form a balanced pair, thus reducing friction 
and wear). The shaft F may be given any number of turns by means 
of the wheel A, at the same time the bevel D may be turned as desired, 
and the problem will be to determine the resulting motion of the bevel 
E. The shaft and cross F here correspond with the arm of the epicylic 
spur-gear trains, for we may assume the bevels locked so that they do 
not turn relative to F, and then give the desired motion to F ; the bevels 
may then be unlocked and, while F remains fixed, such motion may be 
given to either of them as the data require. For example, in Fig. 257 
let A make +5 turns while B makes — 4; to find the resulting motion 
of C. When the bevels are arranged as in Fig. 257 the wheels D and 
E must have the same number of teeth. [It will be found clearer in 
these problems to assume that the motion is positive when the nearer 
side of the wheel moves upward, in which case a downward motion would 
be negative ; or if a downward motion is assumed as positive, then up- 
ward motion would be negative.] Solving the problem by the tabular 
method, we have: 



1° Train locked +5 

2° Train unlocked, arm fixed —9 



E 

+ 5 
+ 9 



F 

+ 5 




3° Resultant motion. 



-4 +14 +5 



Or the wheel C will make 14 turns in the same direction as A. 

Solving this problem by the general equation, and letting D, E, and 
F, in the latter part of the equation, represent the absolute turns of D f 



EPICYCLIC BEVEL TRAINS. 



179 



E, and F respectively, we have 
relative turns E 



-1 



E-F E-5 



relative turns F D—F —4 — 5' 

.*. £= + 14. 

The bevel train in any machine may be driven through some external 
train of gears, in which case the first step must be to determine from the 
given train or trains the absolute 
turns of the respective parts of the 
epicyclic train. For example, in 
Fig. 258 let it be required to find 
the turns of B for + 1 of A . From 
the trains at the left we find that 
+ 1 turn of A will give —2 turns 
to D and + -| turn to F; fronuot 
these we may determine the turns 
of E, c 

-l = 4Pt; .-. E= + 3. 20t 



c 


A 


13 
50. t? 












30 t 


B 


3 






\ 


i i 


/ 


60 t 




. 




II 
F 

II 






















ft 


u 




25 t 


D 


E 








/ 




\ 




50 t 

















Fig. 258. 



7 



-2-r 

+ 3 turns of E will give —6 turns 
to B, or B makes 6 turns in the 
opposite direction to A. 

The bevel train may be a compound train, as shown in Fig. 259, the 

essential difference in this case 
being that the value of the 
bevel train relative to the arm 
is no longer — 1. In Fig. 259 
if B makes —10 turns, and A 
+ 40, find the turns of C. With 
the numbers of teeth as given 
in the figure, we should have 
relative turns D _ 125 . . 28 50 
42 



71 



V 

Fig 



259. 



relative turns E 

50_P-F_^ + 40-F < 
"9 E-F~ -10-F ; 



X 15 9 ; 



'. F=- 



140 
59 



140 
Or, C will turn -— - turns in the same direction as B turns. To check 
59 

this we may use the tabular method: 



180 



AGGREGATE COMBINATIONS. 



° Train locked 

° Train unlocked, arm fixed. . 


D 
140 
59 

- +*« 


E 

140 
59 


F 

140 
59 




° Resultant motion 


. +40 


-10 


140 

59 




170. Other Examples of Epicyclic Trains. — An epicyclic bevel train 
is used in connection with a train containing a pair of cone pulleys, in 

a form of water-wheel governor 
for regulating the supply of 
water to the wheel. Fig. 260 
is a diagram for this train, the 
position of the belt connecting 
7 the cone pulleys being regu- 
lated by a ball governor con- 
necting by levers with the guid- 
ing forks of the belt. The gov- 
ernor is so regulated that when 
running at the mean speed the 
belt will be in its mid-position, 
at which place the turns of E 
and D should be equal, and 
opposite in direction, in which 
case the arm F will not be turning. If the belt moves up from its mid- 
position, and if A turns as shown, the arm F will turn in the same direc- 
tion as the wheel E. As an example, find the diameters x and y if C is tc 
turn downward 1 turn for 25 turns of A in the direction shown ; and must 
the belt be crossed or open? Solving by the general equation, we have 



U~_ y- 



u 



Fig. 260. 



-1 = 



D-F 



( 



25)g + l 



E 



E+l ' 
616 

67.* 
The plus sign indicates that E must turn upward or opposite -to D, 



F 
E = 



and therefore an open belt is required. 



To find the ratio — equate the 

turns of E, found above, to the turns of E for 25 turns of A through 
the belt; thus, 

616 _ 
67 ~~ 
2/. = 308 
" ' x~375' 
Or if the diameter x = 7i inches, then the diameter y would be 6.16 inches. 



x 67 25 ' 



CYLINDER LORING-BARS. 



181 



Epicyclic trains are used, in connection with a screw, in cylinder 
boring-bars to feed the cutter collar along the bar. Let F, Fig. 261, 
represent such a bar, supported in the centres of a lathe. E represents 



tt 



m 



J 



TJMmllJ::. 



:] 



Fig. 261. 

the cutter-holding collar. The cylinder to be bored is made fast to 
the bed of the lathe, and the problem is to feed the collar E through 
the cylinder while E turns with F. To this end E is paired with a 
screw running in a groove in the bar, and driven by a gear D. The 
motion of D is due to the epicyclic train ABCD, the stud on which 
the wheels B and C are mounted being fast to the bar F and so caus- 
ing BC to revolve about the gear A made fast to the tailstock of the 
lathe. 

In Fig. 261 if F turns as shown, and if the screw has four threads per 
inch R.H., find the resulting motion of the collar E for one turn of F. 
Calling the motion of F positive and solving by the tabular method: 



1° Train locked +1 

2° Train unlocked, arm fixed — 1 



3° Resultant motion 







D 

+1 



+i 



F 

+1 




+1 



But the turns of D which we need to determine the motion of E will 
be the turns relative to F, or -{--} — 1= — -f, since the motion of E along 
F is due only to the turns of the screw in F. If the screw makes f of a 
turn in a direction opposite to the motion of F, the collar E will travel 
y 1 /' to the right, which is, therefore, its travel for one turn of the bar. 

It will be evident that, since A is fixed and F turns downward, the 
same result would be obtained by assuming F fixed and turning A up- 
ward once. The train from A to D is arranged so that one gear, C, 
will be a change gear, which makes it possible to vary the rate of motion 
of E. 



182 



AGGREGATE COMBINATIONS. 




Fig. 262. 



Fig. 262 shows an application of the sun and planet wheels, which will 

give the same feed on a boring- 
bar as the arrangement just de- 
scribed. In this case the screw 
S is attached to the collar in the 
same way as before, and the end 
of the screw projects beyond the 
end of the boring-bar and car- 
ries a spur-wheel A. A pinion 
B, supported by an adjustable 
stud in the end of the bar, 
gears with the wheel A. This 
pinion is prevented from turn- 
ing by the slotted link working 
on the stationary pin P, which 
is usually placed much further below the boring-bar than here shown. 
The action of the wheel B is the same as that of C, Fig. 249. 

Let the pitch of the screw S be £" R.H., and let A have 90 teeth and 
B 20 teeth, the bar turning right-handed as seen from the right. Call- 
ing m the turns of B, n the turns of A, a the turns of the bar, and 
relative turns A . . , „ ,, , - - 

^ relative turns £ ' We W m = °> a = + 1 ' and e== ~ i; then ^ == + 1 ^ 
and the nut turns +1; therefore the feed is tXi= T 1 g // , and the collar 
is drawn toward the right. 

To reverse the feed in this machine, the stud of the wheel B is dropped 
so that A and B are no longer in gear; then the idle wheel C is adjusted 
so that it gears with A and B, the stud of the wheel C moving in a T 
slot concentric with the screw S. Now m = 0, a= + 1, and e= + f ; then 
n= + 1 ; and the nut turns + 1 ; therefore the feed is 1 Xi= T V'> 
as before, but now the nut moves toward the left. 

171. Roberts's Winding-on Motion. — On a mule the spun 
yarn is wound upon a slightly tapering spindle in conical 
layers, as shown in Fig. 263, forming what is called a cop. 
The formation of the cop may be divided in two parts: 
1° the forming of the copbottom upon a bare spindle by 
superposing a series of conical layers of yarn with a continu- 
ally increasing vertical angle; and 2° the building of the body 
of the cop by winding the yarn in a series of nearly conical 
layers. If we conceive the winding on to begin at the base 
of the cone, forming the copbottom, it is evident that the 
speed of the spindles must increase in order to wind on the F '„„ 
same amount of yarn for the same travel of the spindle. 

Fig. 264 shows the principal parts of a mule that are concerned in 
the winding on of the yarn. The carriage GG travels in and out 



ROBERTS'S WIND1NG-0N MOTION. 



183 



from the rolls L and carries the spindles T, arranged in a row and driven 

by a long drum and cords, as shown. The cop is shown at K, and the 

position of winding upon the 

cop is governed by a wire, 

called the falter wire, attached 

to the ends of the arms V ', car- o 

ried by a shaft just inside of and 

parallel to the shaft V, which 

itself carries the counter- jailer 

wire V by a series of arms. 

The yarn, in being wound upon 

the cop, passes over the coun- 

ter-faller wire and under the 

faller wire, the shafts of both 

wires being supported by the 

carriage. 

While spinning, the carriage 
moves from a position where 
the spindles are about six inches 
from the rolls L to the right, 
until the drum H of the car- 
riage reaches the position U x \ 
the spindles meanwhile are 
driven by a rope and grooved 
pulley (not shown) attached 
to the long drum shaft. The 
counter-faller wire is then be- 
low the tops of the spindles 
and the faller wire above them, 
the yarn spinning from the tops 
of the spindles. 

The winding-on mechanism 
consists of three parts: 

1° The vibrating arm DD 
(centred upon the stud C, fixed 
to the mule frame), governed 
in its motion by the pinion B, 
whose shaft turns in bearings 
attached to the frame. The 
pinion B is connected to the 
carriage in such a way that they have a constant velocity ratio, and 
the proper relative movement, a rope and drum R being usually employed. 

2° The train HIJ (attached to the carriage), which is made up of a 
chain drum H fixed to a spur gear /, gearing with the pinion J, which 




1S4 AGGREGATE COMBINATIONS. 

pinion is attached to the drum shaft by a ratchet similar to that shown 
in Fig. 199. This allows the drum to turn freely while spinning is going 
on ; and secures the connection of J and the shaft whenever J turns in 
the direction of the arrow. 

3° A chain FF, which connects the drum H with an adjustable block 
E in the quadrant arm. A screw S serves to regulate the position of the 
block in the quadrant arm. This screw may be turned by a wrench 
applied to the squared end D or by means of the bevel gears at C. 

When winding on begins, the carriage is at its extreme right position, 
the drum H being at H 1} and the quadrant in the position D v Now 
if we suppose the quadrant to be fixed in the position D x and allow the 
carriage to move in, the drum H will rotate in the direction of the arrow 
with a very nearly uniform velocity; in this case it will gradually 
accelerate, as the chain is attached to a point not in line with the motion 
of the top of the drum. If, on the other hand, we allow the carriage 
to stand still and swing the quadrant arm from D t to D 2 , the drum 
will rotate in a direction opposite to that of the arrow with a constantly 
decreasing angular velocity proportional to the perpendicular let falL 
from C to the line of the chain, which is the line of connection. Allow- 
ing both of these motions to take place simultaneously, by connecting 
the carriage as described with the pinion B, it will be found that the 
motion of the arm in passing from D^ to D 2 will subtract from the first 
nearly uniform motion of unwinding of the chain a continually decreas- 
ing amount, and thus the spindles, which are driven by the drum H 
through the train and ratchet, will have a constantly accelerating 
angular velocity. 

The faller wire V regulates the position of winding upon the cop, 
and the counter-faller wire V takes up any slack by moving upward, 
the winding on being so planned that only an amount of yarn equal to 
the travel of the carriage is wound upon the cop for each run-in of the 
carriage. 

This motion would be suitable for winding a conical layer of a cer- 
tain size, but a different motion is necessary to form the copbottom. 

As the first layer of the copbottom is placed upon the bare spindle 
(or upon a thin coptube), a nearly uniform motion is called for in the 
winding. This is obtained by placing the block E in the position E 2 , 
the proper one to start the copbottom: the motion of H will now be a 
very gradually accelerating one. As the cop gradually builds up, the 
travel of the faller wire being higher and higher, as shown by the suc- 
cessive layers in Fig. 263, the screw S is turned by means of the bevel 
gears at C, and the position of the nut E is so regulated that the proper 
amount of yarn is wound upon the spindle for each run-in of the carriage. 
The movement of the nut E, outward, is regulated by the counter-faller 
wire V, which throws into gear a train of mechanism operating the 



FUSEE. 185 

screw S whenever the threads W become so taut as to draw the wire 
below its normal position, movement of the nut E away from C causing 
less and less motion to the spindle. 

The motion of the faller wire is governed by an inclined rail, the 
incline being varied so that in building the copbottom the wire will 
move so as to make the winding closer at the bottom, thus increasing 
the diameter more rapidly there than at the top. 

The nut E is moved outward until it reaches a stop which is adjusted 
in position to build the cop upon the copbottom, the winding motion 
being then the same for each run-in of the carriage, the path of the faller 
wire being, however, successively higher and higher. 

The figure shows the carriage running in, the directions of motion of 
the different paths being <shown by arrows. The nut E is now in the 
proper position to build the cop upon the copbottom; for a larger-sized 
cop it would be necessary to allow E to move farther out toward E v 
which is the extreme position. 

172. A Fusee is a contrivance adopted in some of the older forms 
of watches in order to maintain a uniform force upon the train of 
wheels and compensate for the decreasing power of the main-spring. 
In this case the fusee consists of a groove of a helical nature, traced 
upon a conoid, formed by revolving a hyperbola on one of its axes. 
The spring is placed in a cylindrical drum, and this drum is connected 
to the fusee by a cord or chain. As the spring uncoils and its force 
diminishes, the cord being drawn to the cylinder acts on a continually 
increasing arm, the groove being so made that this arm increases in the 
proper ratio. 

In mechanism the fusee is frequently employed to transmit motion, 
and then it enables us to derive a continually increasing or decreasing 
motion from the uniform motion of the fusee shaft. 

The groove of the fusee may be traced upon a cone or other tapering 
surface, or it may be compressed into a flat, spiral curve; in all cases 
the effect produced will be that due to a succession of arms which 
radiate in perpendicular directions from the 
fixed axis, and continually increase or de- 
crease in length. 

Two fusees may be combined (Fig. 265) 
so that the motion produced may be increas- 
ing at first and then decreasing at the last. - 
Such a device, known as a scroll, is used 
to operate the carriage of a spinning-mule, 
which should have an accelerated motion 
up to the middle of its path, and then a 
retarded motion to the end of its path, 
the start and stop being slow. 







CHAPTER XII. 
GEARING.— CONSTRUCTION OF GEAR-TEETH. 

173. In Chapter IV, § 44, it was stated that teeth could be formed 
from rolling cylinders of such shape that by their sliding action the 
same a.v. ratio could be obtained for the axes of the rolling bodies as 
would be obtained if the rolling bodies were assumed to drive each 
other without slipping. 

In order to discuss the action of such gears, and to design them, it 
will be necessary to understand the following definitions of the various 
terms constantly used : 

1° Pitch Surface. — The pitch surface of a toothed wheel, or rack, is 
the elementary surface from which the tooth curves are formed, as 
either one of a pair of rolling cylinders. 

2° Pitch Line. — The pitch line of a gear-wheel is the trace of the 
pitch surface on a plane perpendicular to the axis of the wheel; in a 
cylinder this would be a circle; in a rack where the teeth are formed 
from a plane it would be a straight line. Where the teeth are formed 
from rolling cones the pitch lines are commonly taken as the largest 
intersections of the perpendicular planes, thus giving the bases of the 
rolling cones as their pitch circles. 

3° Pitch Point. — The pitch point of a pair of gear-wheels is the 
point of contact of their pitch lines, as the point c, Fig. 266. The pitch 
point of a tooth is the point where the tooth curve crosses the pitch line, 
as a v Fig. 266. 

4° Pitch. — Diametral Pitch. — The pitch is the distance measured on 
the pitch line from a point on one tooth to the corresponding point on 
the next tooth, as ca v Fig. 266, and is equal to the thickness of the tooth 
plus the space between the teeth, afi t +b x c. In all cases the pitch must 
be an aliquot part of the pitch line, and in order that two wheels may 
gear with each other they must have the same pitch. Thus in Fig. 266 
ca x must equal ca 2 . 

Diametral Pitch. — To lay out the teeth on a pair of wheels, it is 
necessary to use the pitch; but if the pitch is arbitrarily assumed, it 

186 



DIAMETRAL PITCH. 



187 



would usually give awkward decimals in the dimensions of the diam- 
eters. Practically, it is much more important that the diameters should 
be even numbers or convenient fractions, and that the pitch should be 
deduced from the diameter and the number of teeth. The diametral 
pitch is the diameter divided by the number of teeth. Thus, if the gear 
is 10 inches in diameter and has 20 teeth, the diametral pitch is J 
inch. This is commonly expressed by saying that the gear is 2-pitch, 
or 2-P., meaning that it has two teeth corresponding to each inch of 
diameter ; thus a 5-P. gear having 50 teeth would be 10 inches in diam- 
eter. 

Since the circular pitch is the circumference divided by the number 
of teeth, and the diametral pitch is the diameter divided by the num- 
ber of teeth, it follows that 

circular pitch _ circumference _ 
diametral pitch diameter ' 

or, circular pitch = diametral pitch Xx. Thus, a 3-P. gear has a diametral 

pitch of J inch, and a circular pitch of — inches = 1.047 inches. 

5° Backlash. — The backlash is the difference between the space on 
one wheel and the thickness of the' tooth on the other : in Fig. 266 




Fig. 266. 

ajj 2 — afi 1 will be the backlash. In most wheels the thickness of the 
tooth is the same on both the wheels which are in gear, so that the back- 
lash would be the difference between the space and the thickness of 
the tooth, but for constructive reasons the thicknesses may not be 



188 



CONSTRUCTION OF GEAR-TEETH. 



equal, so that the former definition is better. Backlash prevents the 
non-acting sides of the teeth from touching, and some should always be 
provided, but it may be very small in accurately cut gears. 

6° Addendum. — The addendum is the term applied to the length of 
the tooth outside the pitch circle, as d 2 e 2J Fig. 266, and a circle drawn 
through the point e 2 with o 2 as a centre is called the addendum line or 
circle, and limits the tops of the teeth. 

7° Root Circle. — The root line or circle is a line drawn through the 
bottoms of the spaces, as through / 2 , Fig. 266. 

8° Clearance. — The clearance is the difference between the radial 
distance from the pitch line of one wheel to its root circle, and the adden- 
dum of the other wheel, and is the amount by which the tops of the 
teeth of one wheel clear the bottoms of the spaces of the other, as they 
pass the line of centres; thus, in Fig. 266, xy is the clearance on the 
wheel A and is equal to d 2 f 2 —d i e l . In most wheels the addendum is 

the same on each of two 
wheels in gear, so that the 
clearance would be che 
difference between the ra- 
dial distances from the 
pitch circle to the root 
and addendum circles, as 

& 2 \ 2 W 2 6 2 . 

9° Length. — The 
length of a tooth is the 
distance between the ad- 
dendum and root circles 
measured on a radial line, 
as ej 2 , Fig. 266. 

10° Breadth. — The 
breadth of a tooth is the 
distance measured on an 
element of the pitch sur- 
face, between the two 
bounding surfaces of the 
tooth. 

11° Parts of the 
Teeth.— Face and Flank. 
— The face of a tooth is 
that part of the tooth 
curve extending beyond 
the pitch circle, as ca lf 




Fig. 267. 



ca 2 , Fig. 267, and the flank is that part of the curve within the pitch 
circle, as cb 1} cb 2 . If the wheel A is the driver and turning as shown ; it 



ARCS OF APPROACH AND RECESS 



189 



will be seen that the flank of the driver acts upon the face of the 
follower during the approaching action, that is, while the teeth are 
sliding toward each other, and that during the receding action the 
face of the driver will drive the flank of the follower. The acting flank 
is the part of the flank which comes into contact with the face of the 
tooth of the other wheel, for it will be evident from Fig. 267 that the 
entire flank cannot come into contact. Thus the acting flank Of the 
tooth on A is dx = cy, d being the first point on the flank of A to come 
into contact with the face on B. 

12° Point of Contact. — Path of Contact. — Points of contact are 
points where the teeth touch each other, as / and g, Fig. 267, and 
the path of contact is a smooth curve drawn through the successive 
points of contact of a pair of teeth while in action, as the curve dee, 
Fig. 267. The path of contact is always limited by the addendum 
circles, as at d and e, and will be found to always pass through the 
pitch point of the wheels. 

13° Arcs and Angles of Action. — Arcs and Angles of Approach 
and Recess. — In Fig. 268 let C L and D x be a pair of teeth in contact at 




Fig. 268. 



the point d where the contact begins, and let C 2 and D 2 be the same 
pair of teeth in the position where contact just ends at e; then the arc 
of action is the arc through which the pitch line of either wheel moves 
while a pair of teeth are in contact, as af> x = a 2 b 2 . The angles of action 
are the angles through which the wheels move while a pair of teeth 
are in action, and if the diameters are not alike, these angles will be 



190 CONSTRUCTION OF GEAR-TEETH. 

inversely as the radii of the pitch circles, since the arcs which subtend 
them are equal. For A (Fig. 268) the angle of action is a v and for 
B, «,. 

The arcs of approach are a 1 c=a 2 c, the arcs through which the pitch 
lines move while the teeth are moving toward each other, which 
action ends when the points a x and a 2 , the pitch points of the 
teeth, meet each other at c. Similarly the arcs of recess are cb 1 = cb 2 . 
The angles of approach and of recess for the wheel A are the angles sub- 
tended by the arcs of approach and of recess respectively, /? x and y v 
For the wheel B the angles of approach and of recess are /? 2 and y 2 
respectively. 

i In order that one pair of teeth shall not cease their action until the 
next pair are in contact, the arc of action must be at least equal to the 
pitch, and in practice it should be considerably more if much force is 
to be transmitted, so that usually two pair of teeth, or more, are always 
in contact. 

The approaching action is more injurious than the receding action, 
for in approach the friction between the teeth adds to the pres- 
sure on the bearings of the wheels, while in recess the reverse is the 
case. 

14° Line of Action or Line of Connection. — Obliquity of Action. — 
The line of action, or line of connection, is a line normal to the tooth 
curves at their point of contact, as is always the case in pieces in sliding 
contact (see § 111). The component motions along this line will be the 
same for the two points in contact, as will also be the case for the force 
components transmitted. The obliquity of action, or angle of obliquity, is 
the angle which any line of action makes with the common tangent to 
the two pitch lines. Thus in Fig. 268 the teeth C 1 and D t in contact 
at d will have for the line of action the normal dc, and the angle of 
obliquity in that position will be d v The maximum angle of obliquity 
in approach would be the angle of obliquity at the beginning of approach, 
which is d t in the figure, and d 2 would be the maximum angle of obliquity 
in recess, since ce is the line of action at the end of the path of 
contact. 

The angle of obliquity should not be large, as the force required to 
overcome a given resistance would increase if the angle of obliquity 
increased, since the moment arm of the force along the line of connection 
would decrease in the driven wheel if the angle of obliquity increased, 
necessitating a greater pressure by the driving tooth. 

174. Gearing Classified. — The following are the varieties of tooth 
wheels commonly met with in practice. 

1° Spur Gearing (Fig. 269). — Here the axes of the wheels are parallel. 
If we suppose the number of teeth to be increased indefinitely, their 



GEARING CLASSIFIED. 



191 





size being at the same time correspondingly diminished, they will 

finally become mere lines, or elements of 

surfaces in contact, thus giving the figure 

at the right, which figure shows the pitch 

surfaces. 

If the teeth of one of the wheels are replaced 
by pins, and the teeth of the other are made to 
work properly with the pins, we have what is 
called pin gearing. 

Should the radius of one of the wheels be made infinite, the pitch line 
becomes a straight line, and we have a rack. 

2° Bevel Gearing (Fig. 270). — Here the axes intersect, and the pitch 
surfaces are cones, having a common apex at the point of intersection 
of the axes. When the axes intersect at right angles and the wheels 
are equal, the gears are called mitre-wheels. 



Fig. 269. 





Fig. 270. 



Fig. 271. 



3° Skew Gearing (Fig. 271).— Here the axes lie in different planes, 
and the pitch surfaces are hyperboloids of revolution. This class of 
gearing is not very generally used, owing to the difficulty of forming 
the teeth, it being possible, in most cases, to make the connection by 
means of two sets of bevel gears. 

In cases 1°, 2°, and 3° the teeth touch each other along straight 
lines, parallel to the axes in 1°, passing through the apexes of the pitch 
cones in 2°, and approximating in their general direction to the common 
element of the two hyperboloids in 3°. 

4° Twisted Gearing (Fig. 272).— If we suppose one of a pair of 
engaging circular wheels, belonging to either of the first three classes, 
to be uniformly twisted on its axis, each suc- 
cessive transverse plane being rotated through 
a greater angle, the other wheel receiving a cor- 
responding twist as shown, we shall have a case 
of twisted gearing. 

The wheels thus formed will gear together as 
well as before. The teeth have faces of a heli- 
coidal nature, and by increasing the number of 
teeth indefinitely, helical lines or elements would 




Fig. 272. 



result, giving the same pitch surface as before. 



192 CONSTRUCTION OF GEAR-TEETH. 

In the case shown, pressure will result along the axes of the gears. 
To neutralize this axial pressure, the twisting can be made to start at 
the central plane of the wheel, and proceed the same on each side, as 
shown at A. 

5° Screw Gearing (Fig. 273). — Here the teeth also have a helicoidal 
form, as in twisted gearing, and reduce to helical lines ; but these helices 
lie upon cylinders whose axes are in different planes, the pitch surfaces 
touching in a single point only. As illustrated by the "worm and 
wheel/' it is the screw-like action alone of one wheel on the other which 
transmits the motion. 







uwuuuuuuuu 



■=( 



fb 




Fig. 273. Fig. 274. 

6° Face Gearing (Fig. 274) is not much used in modern machinery. 
The teeth generally consist of turned pins projecting from circular discs, 
but may be arranged on other surfaces than planes; the axes also may 
be inclined to each other. 

In this case the teeth are circular in section, and touch only in one 
point; and when the number is increased indefinitely, two circles touch- 
ing each other at their circumference will result. Face-wheels, then, 
have no pitch surfaces properly so called, but surfaces of some kind 
are required to support the teeth. This class of gearing was best 
adapted to wooden mill machinery, and at one time was used almost 
exclusively for that purpose. 

175. Fundamental Law Governing the Shapes of Curves Suitable for 
Tooth Curves. — In Fig. 275 let A and B be two pieces in sliding contact 
at the point d, with nn as the common normal to the acting surfaces at 
d, and assume the piece B, turning R.H., to be driving the piece A. 
Draw o t a and o 2 b perpendicular to nn. The direction of motion of the 
point a in A, and also the direction of motion of b in B, are, in this posi- 
tion, along nn; therefore the l.v. of a must be equal to the l.v. of b, for 
if l.v. of a were greater than l.v. of b, the curves in contact at d would 
separate. This fact is also evident by noticing that nn is the line of 
connection between the sliding pieces in contact at d, and the com- 
ponents along nn of the l.v's of any points in either A or B situated on 
nn must be equal; a and b, which are, in the given position, moving 
along nn, must therefore have their l.v's equal. 

l.v. a = a.v. AXo^a and l.v. b = a. v. BXo 2 b. 
But l.v. a = l.v. b; 

.'. a.v. A : a.v. B = ojb : o x a. 



LAW GOVERNING SHAPES OF TOOTH CURVES. 193 

But by construction the triangles o x ac and o 2 bc are similar, from which 

ojb : o x a = o 2 c : o x c; 
.*. a.v. A : a.v. B = o 2 c : o x c. 

If now we draw the two circles shown dotted through the point c with 
the centres o x and o 2 , and assume the circles to move in rolling contact, 
we should have 

a.v. o t : a.v. o 2 = o 2 c : o x c. 

Therefore the two pieces A and B, with their axes at o x and o 2 respect- 
ively, have an a.v. ratio, due to their sliding contact, exactly the same 




Fig. 275. 

at this instant as that of two rolling cylinders on the same axes and in 
contact at c. Thus if a constant a.v. ratio is to be maintained, which is 
the special function of gearing, it is only necessary that the tooth curves 
shall be so shaped that at any point of contact the common normal to the 
curves shall pass through the pitch point of the wheels. For if, on moving 
the pieces A and B into some other position, and drawing the normal to 
the acting curves at the new point of contact, it were found that this 
normal passed through some other point than c on the line of centres, 
it could be proved that the a.v. ratio was the same as that of some 
other pair of rolling cylinders in contact at the point where the new 
normal crosses the line of centres. 

This Law, that the normal to the tooth curves at any point of contact 
■must pass through the pitch point of the gears, is fundamental to all types 
of gearing if constant a.v. ratio is to be obtained. 

Fig. 276 shows a pinion and annular wheel with one pair of teeth in 

contact at d, nn being the common normal to the curves at d. o x a and 

o 2 b are drawn perpendicular to nn. Then, bv the same reasoning as in 

Fig. 275, 

l.v. a = l.v. b; 

l.v. a = a.v. o x Xo x a and l.v. 6 = a.v. o 2 Xo 2 b; 
.*. a.v. o x : a.v. o 2 = o 2 b : o x a = o 2 c : o x c; 



194 CONSTRUCTION OF GEAR-TEETH. 

or, the teeth in contact at d would give to the axes o t and o 2 the same 
a.v. ratio as that of the two cylinders in internal contact at c. 



0i 



-■^-^tps 


m 


___, "' at 

n 


//Y 
/ 


Fig. 276. 





176. Rate of Sliding. — To determine the rate of sliding of one tooth 
upon another at any position it will be necessary to find the l.v. of the 
point of contact d in each of the teeth, and resolve these l.v's into their 
components along the common normal and the common tangent. In 
Fig. 276 let de represent the l.v. of d around o t ; the components of de 
along the common normal and tangent are df and dg respectively. The 
direction of the motion of d around o 2 is along the line dh. To find the 
magnitude of its l.v. we have df as its component along the common 
normal, since this normal is the line of connection between the two 
sliding surfaces, and components along the line of connection must be 
equal. This will give dh as the l.v. of d around o 2 , and dk as its com- 
ponent along the common tangent. The rate of sliding will be found to 
be gk, equal to dg + dk, since the components along the tangent act in 
opposite directions. 

177. General Problem. — Conjugate Curves. — Given the face or flank 
of a tooth of one of a pair of wheels, to find the flank or face of a tooth 
of the other. The solution of this problem depends on the funda- 
mental law, § 175. In Fig. 277 let the flank and face of a tooth on 
A be given. If A is considered as the driver, points on the flank, as 
a and b, will be points of contact in the approaching action, and by the 
law they can properly be points of contact only when the normals to 
the flank at these points pass through the pitch point: therefore drawing 



GENERAL PROBLEM. 



195 



ac and bd normals to the flank from the points a and b respectively, and 
then turning A backward until 
the points c and d are at the 
pitch point, we find positions a x 
and b x which a and b respect- 
ively must occupy when they 
can be points of contact with 
the face of a tooth of the other 
wheel. The point a x must be a 
point on the desired face of a 
tooth on the wheel B when the 
pitch circles have been moved 
backward an arc equal to c x c, 
that is, so that c is at the 
pitch point. To find this 
point when the teeth are in 
the original position, it is 
necessary to move the wheels 
forward, the wheel B carry- 
ing with it the point a x and 
the normal a x c x until the point 
c x has moved through an arc 
c x c 2 equal to c x c; this will 
carry the point a x to a 2 , and 
the normal a x c x to a 2 c 2 . During 
this same forward motion the 
normal a x c x moving with the 
wheel A will return to its orig- 
inal position ac. 

In a similar manner the point b x , which can be a point of contact of 
the given flank with the desired face, is a point on this face when the 
pitch circles of the wheels are moved backward an arc equal to c x d. Mov- 
ing them forward the same distance, the point b x and normal b x c v moving 
with the wheel B, will be found at b 2 and b 2 d 2 . This process may be con- 
tinued for as many points as may be needed to give a smooth curve. 
The curve drawn through the points a 2 b 2 c x will be the required face. 

A similar process gives the flank of the tooth on the wheel B which 
will work properly with the given face. The normals taken in the figure 
are eg and fh, the positions of e and / when they can be points of contact 
being e x and f x ; and the points on the required flank when in the original 
position are e 2 and f 2 . 

A smooth curve passed through the points of contact a x b x c x e x f x will 
be the path of contact, the beginning and end of which will be deter- 
mined by the addendum circles of B and of A respectively. 




Fig. 277. 



196 



CONSTRUCTION OF GEAR-TEETH. 



Another method of solving the above problem is shown in Fig. 278, 
where o x and o 2 are a pair of plates whose edges are shaped to arcs of the 
given pitch circles AA X and BB lf due allowance 
being made for a thin strip of metal, gh, connect- 
ing the plates, to insure no slipping of their edges 
on each other. 

Attach to o 2 a thin piece of sheet metal, M, the 
edge of which is shaped to the given curve aa t ; 
and to o x a piece of paper, D, the piece M being 
elevated above o x to allow space for the free move- 
ment of D. Now roll the plates together, keeping 
the metallic strip gh in tension, and, with a fine 
marking-point, trace upon the paper D, for a suffi- 
cient number of positions, the outline of the curve aa v A curve just 
touching all the successive outlines on i), as ee lf is the corresponding 
tooth curve for o x . 

Conjugate Curves. — Any two curves so related that, by their sliding 
contact, motion will be transmitted with a constant a.v. ratio, as in roll- 
ing cylinders, are called conjugate curves. The curves, in this case, are 
often called odontoids. 

178. After finding the proper shapes for the tooth curves and know- 
ing the pitch, backlash, addendum, and clearance, the teeth may be 
draw T n as in Fig. 279. A convenient method of laying off the pitch ab x 




Fig. 278. 




Fig. 279. 

on the pitch circle is indicated in the figure; let ab be equal to the pitch, 
laid off from the pitch point, on the tangent; starting from b space off 



THE DRAWING OF ROLLED CURVES. 



197 



toward a equal divisions sufficiently small so that when spaced back on 
the pitch circle the difference between chord and arc may be neglected; 
one of these divisions, as c, will come sufficiently near to a that it may 
be considered to be on the pitch circle as well as on the tangent. From 
this point c space back on the pitch circle the same number of divisions 
giving the point b lf and ab x will be very nearly equal to ab. This method 
of spacing may be used to construct the tooth curves in the two sys- 
tems of gearing to be discussed in this chapter. 

If the flanks are extended until they join the root line, a very weak 
tooth would often result; to avoid this, a fillet is used which is limited 
by the arc of a circle connecting the root line with the flank, and lying 
outside the actual path of the end of the face of the other wheel. This 
actual path of the end of the face is called the true clearing curve, the 
construction of which is taken up later in § 179, 5°. 

Fig. 279 shows a rack and pinion in gear. If the pinion drives, the 
path of contact will be kal. The ends of the acting flanks are also indi- 
cated at m and n. 

179. The Drawing of Rolled Curves. — Any curve described by a 
point carried by one line which rolls upon another may be called a rolled 
curve. 

The line which carries the tracing-point is called the generatrix, de- 
scribing line, or describing circle, while the one in contact with which it 
rolls is called the directrix or base line; either line may be straight, or 
both may be curved. 

1° The Cycloid. — This curve is traced by a point in the circumfer- 
ence of a circle which rolls upon a straight line as a directrix, as the 
curve aa 2 a 3 (Fig. 280) traced by the point a in the circumference of the 
circle which rolls on the straight tangent line ae. 




b c $ 
Fig. 280. 



To construct the curve, lay off on the circumference of the rolling 
circle a series of equal divisions as shown, and on the tangent line a 
similar series of divisions such that the divisions on the arc are of the 
same length as those on the straight line, and that the corresponding 
points will roll into each other. This division may be made by carefully 



198 



CONSTRUCTION OF GEAR-TEETH. 



using spacing-dividers, setting them so fine that the difference between 
a chord and its arc is inappreciable, or by rectifying by calculation a 
portion of the arc of the rolling circle and dividing both into the same 
number of equal parts, as has been done in the figure. The centre of 
the rolling circle moves on a line oo±, parallel to the tangent line, and its 
position can be found for each division by erecting perpendiculars 
through the divisions on the directrix. To find a point, as a 4 , corre- 
sponding to the fourth division: Here the centre of the rolling circle is 
at o 4 , and the circle is tangent to the directrix at e (perpendicularly 
under o 4 ), the point e t on the circle having rolled to the point e on the 
directrix. Hence striking an arc from o 4 with a radius o 4 e, equal that 
of the describing circle, and intersecting this with an arc struck from 
the tangent point e with a radius equal to the cord ae lf we obtain the 
point a 4 . In the same way other points may be found. The instan- 
taneous centre of the rolling circle is at the tangent point e, and a 4 e is 
the normal, its length representing the radius of curvature of the cycloid 
at the point a 4 . In fact the cycloid might be drawn by making it tan- 
gent to a series of arcs struck as above from the divisions of the directrix. 

Another method of locating the point a 4 is to draw through the point 
e x on the circle a line e t a 4 parallel to the directrix and noting where it 
intersects the corresponding arc of the rolling circle. 

2° The Epicycloid. — This curve is traced by a point in the circum- 




Fig. 281. 



ference of a circle which rolls on the outside of another circle as a direc- 
trix. 

The construction, Fig. 281, is similar to that of the cycloid, and the 
figure is lettered to correspond; the centre of the describing circle moves 



THE DRAWING OF ROLLED CURVES 



199 



on the arc oo x o A ; the successive points of tangency are b, c, d, and the 
points b lf c 1} d v roll to these points, the point a moving to the points 
a v a 2> a iJ successively. 

If the describing circle is larger and rolls internally upon the direct- 
ing circle, an epicycloid will still be rolled. Fig. 282 shows the epi- 
cycloid aa x a z so rolled, the large circle ae 2 f 2 rolling internally on the 
circle aef. If, as in Fig. 282, the diameter of the describing circle ae 2 f 2 
rolling internally is equal to the sum of the diameters of the directing 




Fig. 282. 



circle and the small describing circle ab x c x , the two describing circles will 
trace exactly the same epicycloid, as shown by the points a 2 , a 4 , found 
by the small circle, and a v a 3 , a 5 , found by the large circle. This double 
generation of the epicycloid will be referred to in the discussion of the 
teeth of annular wheels using these curves. 

3° The Hypocycloid. — This curve is traced by a point in the circum- 
ference of a circle rolling inside of another circle. 



200 



CONSTRUCTION OF GEAR-TEETH. 



Fig. 283 shows the construction of a hypocycloid, the letters corre- 
sponding to those in the previous curves; the small circle ab^ rolling 
to the right inside of the large circle, abed, traces the curve aa 2 a 3 a 5 . If 
another circle, ae 2 f 2 , whose diameter is equal to the difference between the 
diameters of the directing circle and the describing circle ab^, as in Fig. 
283, is rolled inside the directing circle, exactly the same hypocycloid 
will be traced as that traced by the circle ab x c ly and these hypocycloids 



r\ 




Fig. 283. 



will coincide as shown in the figure, provided the describing circles roll 
in opposite directions; thus in the figure the circle aej 2 rolls to the left 
and gives the points a i} a 4 , a 6 , while the circle ab l c l rolls to the right and 
gives the points a 2 , a 3 , a 5 . This double generation of the hypocycloid will 
be referred to in discussing annular wheels using these curves. 

When the diameter of the rolling circle is one-half that of the circle 
in which it rolls, the hypocycloid becomes a diameter of the directing 
circle. 

4° The Involute of the Circle. — This curve may be considered as 
generated by a point in a straight line which rolls upon a circle. It may 
also be regarded as generated by unwinding an inextensible fine thread 
from a cylinder; the thread being always taut and always tangent to 
the cylinder, its length is thus equal to the arc from which it was unwound. 



THE DRAWING OF ROLLED CURVES. 



201 




Fig. 284. 



To find one point, as a 2 , in the involute (Fig. 284) : Draw the radius 
oc, and perpendicular to it draw the 
tangent ca 2 . Make the tangent ca 2 
equal in length to the arc ac, and a 2 is 
a point on the involute. The radius 
of curvature of the involute at a 2 is a 2 c. 
It is generally most convenient to lay 
off equal divisions on the directing 
circle, and on some straight line a 
similar series of divisions equal to those 
of the circle. Then draw tangents at 
the divisions of the directrix, and on 
these lay off distances corresponding 
to the different arcs, the distances being 
taken from the divided straight line. 
From the construction it can be seen 
that the normal at any part of the 
involute is always tangent to the 
directing circle. In gearing where in- 
volute curves are used for the teeth, 
the directing circle is called the base circle. 

5° Epitrochoid. — The term epitrochoid is used in a general way to 
name those curves traced by rolling one circle upon another when the 
marking-point is not situated upon the circumference of the rolling 
circle. When the marking-point is situated outside of the circumference 
of the rolling circle, the looped curve traced is known as a curtate epi- 
trochoid; when it is situated inside, the curve is known as a prolate 
epitrochoid. 

The above method of drawing the cycloidal curves may here be 
used; but the tracing-point, now not being on the circumference of the 
rolling circle, must be located in its different positions by a method of 
triangulation from points whose positions are known. 

In Fig. 285 the circle A with its centre at o rolls on the line B. The 
point a, carried by A, will trace an epitrochoid. To draw the curve 
first lay off equal spaces on A and B, pb 1 = pb, pc x = pc, such that b rolls 
to b v c to c v etc., and find the corresponding positions of the centre of 
A, o^ etc. To locate one point on the curve, as a 3 , when d rolls to d lf 
we know that a is always situated a distance oa from the centre of the 
rolling circle, and that when d rolls to d x the centre is at o 3 ; therefore 
a must be somewhere on an arc struck from o 3 with a radius o 3 a 3 = oa; 
also a is distant da from d, and when d rolls to d 1 the point a must be 
found at a 3 on the arc about o 3 distant d x a 3 =da from d v As d t is the 
instantaneous centre, a is moving about it for the instant. In the 
same manner other points, as a v a 2 , may be found. 



202 



CONSTRUCTION OF GEAR-TEETH. 



If the arcs drawn from b x , c 1} d x , etc., are extended, it will be found 
that the epi trochoid is tangent internally to them, and the curve may 
be drawn without finding the points a lf a 2 , etc., as follows: After laying 
out the points which roll into each other, as b, b lf etc., with b x as a centre 
and ab as a radius draw an arc a t ; with c t as a centre and ac as radius 
draw arc a 2 , etc. ; then draw the curve E internally tangent to these arcs. 

The clearing curve defined in § 178 is an epitrochoid, and is constructed 



' i 1 1 h- 

i o 2 /o 3 o 4 o s 6 




Fig. 285. 



by the method just described, as will be evident on referring to Fig. 279, 
where the construction is clearly shown. 

180. Spur Gearing, Cycloidal System. — Generation of the Tooth 
Outline. — In Fig. 286 let o x and o 2 be the centres of the two wheels A 
and B, their pitch circles being in contact at the point a. Let the 
smaller circles C and D, with centres at p 1 and p 2 , be placed so that they 
are tangent to the pitch circles at a. Assume the centres of these four 
circles to be fixed and that they turn in rolling contact; then if the 
point a on the circle A moves to a lt a 2 , a 3 , the same point on B will move 
to 6„ b 2 , b 3 , and on C to c v c 2 , c 3 . Now if the point a on the circle C 
carries a marking-point, in its motion to c t it will have traced from the 
circle A the hypocycloid ajC lf and at the same time from the circle B 
the epicycloid Vi- This can be seen to be true if the circles A and B 
are now fixed; and if C rolls in A, the point c t will roll to a 1} tracing the 



CYCLOIDAL SYSTEM. 



203 



hypocycloid c^; while if C rolls on B, q will trace the epicycloid c t b v 
These two curves in contact at q fulfil the fundamental law for tooth 
curves, that the normal to the two curves at the point c l must pass through 
a; as C starts to roll on either A or B, a is the instantaneous centre of 
C and therefore the direction of motion of c 1} and so the tangent to the 
curves at c t are perpendicular to the radius ac x to the instantaneous 
centre. Similarly, if the original motion of the circles had been to a 2 , 
b 2 , c 2 , the same curves would be generated, only they would be longer 
and in contact at c 2 . If the hypocycloid c 2 a 2 is taken for the flank of a 




Fig. 286. 



tooth on A, and the epicycloid c 2 b 2 for the face of a tooth on B, and if 
c 2 a 2 drives c 2 b 2 toward a, it is evident that these two curves by their 
sliding action, as they approach the line of centres, will give the same 
type of motion to the circles as the circles had in generating the curves, 
which was pure rolling contact. Therefore the two cycloidal curves 
rolled simultaneously by the describing circle C will cause by their 
sliding contact the same a. v. ratio of A and B as would be obtained by 
A and B moving with pure rolling contact. 

If now the circles A, B, and D are rolled in the opposite direction to 
that taken for A, B, and C, and if the point a moves to a 4 , b 4 , and c? t on 
the respective circles, the point a on D while moving to d 1 will trace 
from A the epicycloid a 4 d 1} and from B the hypocycloid b 4 d v The curve 



204 



CONSTRUCTION OF GEAR-TEETH. 



a 4 d t may be the face of a tooth on A, and b 4 d 1 the flank of a tooth on B, 
the normal d t a to the two curves in contact at d x passing through a. 
The flank and face for the teeth on A and B, respectively, which were 
previously found have been added to the face and flank just found, 
giving the complete outlines, in contact at d v 

If now the wheel B is turned L.H., the tooth shown on it will drive 
the tooth on A, giving a constant a. v. ratio between A and B until the 
face of the tooth on B has come to the end of its action with the flank 
which it is driving, at about the point c 2 . 

The following facts will be evident from the foregoing discussion: 
in the cycloidal system of gearing, the flank and face which are to act upon 
each other must be generated by the same describing circle, but the describ- 
ing circles for the face and flank of the teeth of one wheel need not be 
alike. The path of contact is always on the describing circles; in Fig. 286 
it is along the line d x ac v See also § 189. 

181. Interchangeable Wheels. — A set of wheels any two of which 
will gear together are called interchangeable wheels. For these the 
same describing circle must be used in generating all the faces and 
flanks. The size of the describing circle depends on the properties of 
the hypocycloid, which curve forms the flanks of the teeth (excepting 
in an annular wheel). If the diameter of the describing circle is half 
that of the pitch circle, the flanks will be radial (Fig. 287, A), which 




Fig. 287. 



gives a comparatively weak tooth at the root. If the describing circle 
is made smaller, the hypoc}^cloid curves away from the radius (Fig. 287, B) 
and will give a strong form of tooth; but if the describing circle is larger, 
the hypocycloid will curve the other way, passing inside the radial 
lines (Fig. 287, C) and giving a still weaker form of tooth, and a form 
of tooth which may be impossible to shape with a milling-cutter. 



INTERCHANGEABLE WHEELS. 



205 



From the above the practical conclusion would appear to be that the 
diameter of the describing circle should not be more than one-half that 
of the pitch circle of the smallest wheel of the set. It will be found, 
however, that when the diameter of the describing circle is taken five- 
eighths the diameter of the pitch circle, the curvature of the flanks will 
not be so great, with the ordinary proportions of height to thickness of 
teeth, that the spaces are any wider at the bottom than at the pitch 
circle: this being the case, the teeth can be shaped by a milling-cutter. 

In one set of wheels in common use the diameter of the describing 
circle is taken such that it will give radial flanks on a 15-tooth pinion, 
or five-eighths that of a 12-tooth pinion, the smallest wheel of the set. 
This describing circle has been used with excellent results. 

As an example, given an interchangeable set of cycloidal gears, 2-P., 
radial flanks on a 15-tooth pinion; a pinion having 24 teeth is to drive 
one having 30 teeth. The diameter of a 2-P., 15-tooth pinion would be 
1\ inches ; to give radial flanks on this pinion the diameter of the describ- 




Fig. 288. 

ing circle would be 3| inches. This is the diameter of the describing- 
circle for all the faces and flanks for any gear of the set. The 24-tooth 
pinion will have a diameter of 12", and the 30-tooth will have 15" diame- 
ter. This will give the diagram in Fig. 288, ready for the rolling of the 
tooth outlines. 

182. To draw the teeth for a pair of cycloidal wheels, and to deter- 
mine the path of contact. — In Fig. 289, given the pitch circles A and B 
and the describing circles C and D, C to roll the faces for B and the 



206 



CONSTRUCTION OF GEAR-TEETH. 



flanks for A, while D is to roll the faces for A and the flanks for B. These 
curves may be rolled at any convenient place. In the figure, if the 
wheel A is to be the driver and is to turn as shown, any point, as b, on A 
may be chosen, and a point a on B at a distance from the pitch point 




Fig. 289. 



af=bf. The epicycloid and hypocycloid rolled from a and b respectively, 
and shown in contact at b 2 , would be suitable for the faces of the teeth 
on B and the flanks of the teeth on A respectively, and could be in action 
during approach. The curves may be rolled as indicated by the light 
lines. The method used to roll these curves is shown in Fig. 290, 
where the circle C is tracing a hypocycloid on A from the point 
o. Assume the circle C to start tangent to A at o and to roll as 
shown, drawing it in as many positions as may be desired to 
obtain a smooth curve, and these positions do not need to be equi- 
distant; thus in the figure the centre of C is at b, c, and d for 



LIMITS OF THE PATH OF CONTACT. 



207 



the three positions used. Since the circle C rolls on A, the dis- 
tance measured on A from o to a tangent point of C and A is equal 
to the distance measured on C from that tangent point to the hypo- 
cycloid. The method of spacing off these equal arcs for the successive 
positions is the same as described in § 178, Fig. 279, for laying off 
the pitch on the pitch circle, and is clearly indicated in Fig. 290. 




Returning to Fig. 289, the circle D is to roll the faces for the teeth 
on A and the flanks for the teeth on B. These curves may also be 
rolled from any convenient points, as c and d equidistant from /. The 
face thus found from A may be traced and then transferred to the flank 
already found for the teeth on A at the point b, giving the curve b 2 bc' , 
the entire acting side of a tooth on A. Similarly by transferring the 
flank dd s to the point a we have b 2 ad' , the shape of the teeth for the wheel 
B. It will be seen that the face on A could have been rolled from b 
as well as from c, so that the entire tooth curve could be rolled from b, 
and similarly the other tooth curve could have been rolled from the 
point a. After finding the tooth curves, and knowing the addendum, 
clearance, and backlash, the teeth may be drawn. In Fig. 289 the teeth 
are drawn without backlash, and in contact on their acting surfaces at 
h and k. The path of contact is efg on the describing circles and is lim- 
ited by the addendum circles. 

183. Limits of the Path of Contact. — Possibility of any desired 
Action. — If, in Fig. 289, the teeth of either wheel are made longer, the 
path of contact and arc of action are increased; the extreme limit of 
the path of contact would therefore be when the teeth become pointed. 

It is often desirable to find whether a desired arc of action in approach 
or in recess may be obtained before rolling the tooth curves. Given 



208 



CONSTRUCTION OF GEAR-TEETH. 



the pinion A driving the rack B as shown in Fig. 291 ; to determine if 
an arc of approach equal to ab is possible. The path of contact must 
then begin at c, where the arc ac is equal to ab. The face of the rack's 
tooth must be long enough to reach from b to c, and this depends on the 
thickness of the tooth measured on the pitch line, since the non-acting 
side of the tooth must not cause the tooth to become pointed before 
the point c is reached. To see if this is possible without the tooth curves, 
draw a line from c parallel to the line of centres (in general this line is 
drawn to the centre of the wheel ; the rack's centre being at infinity gives 




the line parallel to the line of centres), and note the point d where this 
line crosses the pitch line of the rack. If bd were just equal to one-half 
the thickness of the tooth, the tooth would be pointed at c, and the 
desired arc of approach would be just possible; if bd were less than one- 
half the thickness of the tooth, the tooth would not become pointed 
until some point beyond c was reached, so that the action would be 
possible and the teeth not pointed, as shown by the figure. 

If it is desired to have the arc of recess equal to the arc af, then the 
path of contact must go to g, and the face of the pinion must remain in 
contact with the flank of the rack until that point is reached, or the 
face must be long enough to reach from / to g. Drawing a line from g 
to the centre of the pinion A, we find that the distance fh is greater 



ANNULAR WHEELS. 



209 



than one-half of fk, which is taken as the thickness of the tooth ; there- 
fore the desired arc of recess is not possible even with pointed teeth. 

184. Annular Wheels. — Fig. 292 shows a pinion A driving an annular 
wheel B, the describing circle C generating the flanks of A and the faces 
of B, which in an annular wheel lie inside the pitch circle, while D generates 
the faces of A and the flanks of B. The describing circle C is called the 
interior describing circle, and D is called the exterior describing circle. 
The method of rolling the tooth curves, and the action of the teeth, are 
the same as in the case of two external wheels, the path, of contact being 

• - (b 




.Fig. 292. 

in this case efg when the pinion turns R.H. If these wheels were of an 
interchangeable set, the describing circles would be alike and found as 
explained in § 181, and the annular would then gear with any wheel of 
the set excepting for a limitation which is discussed in the following 
paragraph. 

185. Limitation in the Use of an Annular Wheel of the Cycloidal 
System. — Referring to Fig. 292, it will be evident that, if the pinion 



210 CONSTRUCTION OF GEAR-TEETH. 

drives, the jaces of the pinion and annular will tend to be rather near each 
other during recess (during approach also on the non-acting side of the 
teeth). The usual conditions are such that the faces do not touch; but 
the conditions may be such that the faces will touch each other without 
interference, for a certain arc of recess; or, finally, the conditions may 
be such that the faces would interfere, which would make the wheels 
impossible. 

To determine whether a given case is possible it is necessary to refer 
to the double generation of the epicycloid and of the hypocycloid, § 179, 
2° and 3°. The acting face of the pinion, Fig. 292, is rolled by the ex- 
terior describing circle D, while the acting face of the annular is gener- 
ated by the interior describing circle C. Two such faces are shown in 
Fig. 293 as they would appear if rolled from the points g and h, equi- 
distant from the pitch point k. The acting face of A is an epicycloid, 
and is made by rolling the circle D to the right on A; in § 179, 2°, it 
was seen that a circle whose diameter is equal to the sum of the diam- 
eters of A and B would roll the same epicycloid if rolled in the same 
direction. This circle is E, Fig. 293, and is called the intermediate de- 
scribing circle of the pinion. The acting face of the annular is a hypo- 
cycloid rolled by the interior describing circle C rolling to the left inside 
of B; in § 179, 3°, it was seen that the same hypocycloid would be rolled 
by a circle whose diameter is equal to the difference between the diam- 
eters of B and C, provided it is rolled in the opposite direction. This circle 
is F, Fig. 293, and is called the intermediate describing circle of the 
annular. 

If now the four circles A, B, E, and F turn in rolling contact, through 
arcs each equal to kg, the point k will be found at g, h, m, and n on the 
respective circles, the point k on E having rolled the epicycloid gm, 
while k on F rolls the hypocycloid hn. 

To determine whether these faces do or do not touch or conflict, 
assume that the given conditions gave the circles E and F coincident 
as in Fig. 294, where 

diam. A + diam. D = diam. £' = diam. B — diam. C = diam. F. 

Here if the three circles A, B, and (EF) turn in rolling contact, the 
point k moving to g on A will move to h on B and to (mri) on the com- 
mon intermediate circle. This means that the common intermediate 
circle could simultaneously generate the two faces; therefore the two 
faces are in perfect contact on the intermediate circle. This contact 
would continue until the addendum circle of one of the wheels crosses 
the intermediate circle, the addendum circle crossing first necessarily 
limiting the path of contact. 

The above may be stated as follows: 7/ the intermediate describing 



LIMITATION IN THE USE OF AN ANNULAR WHEEL. 211 



circles of the pinion and annular coincide, the faces will be in contact in 
recess, if the pinion drives, in addition to the regular path of contact. 

If in Fig. 294 the exterior describing circle, for example, should be 
made smaller, as in Fig. 295, then the intermediate of the pinion would 




Fig. 293. 

be smaller than that of the annular; but if the exterior describing circle 
is smaller, the face gm will have a greater curvature and Avill evidently 
curve away from the face hn, so that no contact between the faces can 
occur, as is shown in Fig. 295. Here no additional path of contact 



212 



CONSTRUCTION OF GEAR-TEETH. 



occurs, and it is evident, if the arcs kg, km, kn, and kh are equal as they 
must be, if the circles move in rolling contact, that the smaller D be- 




Fig. 294. 




Fig. 295. 

comes (and consequently E) the greater will be the space between the 
faces. 



LIMITATION IN THE USE OF AN ANNULAR WHEEL. 213 

This may be stated as follows: // the intermediate describing circle of 
the pinion is smaller than that of the annular, the faces do not touch, and 
the action is in all respects similar to the cases of external wheels. 

In Fig. 296 the exterior describing circle D is made larger than it is 




Fig. 296. 



in Fig. 294. so that the intermediate E of the pinion is larger than F, 
that of the annular. Making the circle D larger would give the face of 
the pinion less curvature, which would cause the curve gm to cross the 
curve hn, giving an impossible case. Therefore, if the intermediate of 
the pinion is greater than that of the annular, the action is impossible. 

186. To find the smallest annular to gear with a given pinion. — This 
will be most clearly shown by solving a problem. Let the data be as 
follows: Cycloidal, interchangeable gears, 8-P., radial flanks on a 12-tooth 
pinion ; find the smallest annular which a 20-tooth pinion can drive, and 
show the path of contact if the pinion turns R.H. The successive steps 
are shown in Fig. 297. The exterior and interior describing circles are 
f" diameter, the gears being 8-P., with radial flanks on a 12-tooth pinion. 
The diameter of the 20-tooth pinion is - 2 / = 2y ; and the diameter of the 
intermediate of the pinion will be 2^+f = 3y. The smallest annular 
will be of such size that its intermediate circle will coincide with that 
of the pinion; but the diameter of the intermediate of the annular is 



214 



CONSTRUCTION OF GEAR-TEETH. 



equal to the difference between the diameters of the annular and of the 
interior describing circle, or 

diam. of annular — f" = 3J" ; 
.*. diam. of annular = 3J+} = 4", 

and the smallest annular will have 32 teeth. 

The path of contact, limited by the addendum circles drawn in Fig. 



h— -4 Ae 

\ < / 3H 




Fig. 297. 

297 is abc, the usual path, and in addition the path bd on the common 
intermediate. 

187, Low-numbered Pinions, Cycloidal System. — The obliquity of 
action in cycloidal gears is constantly varying ; it diminishes during the 
approach, becoming zero at the pitch point, and then increases during 
the recess. For wheels doing heavy work it has been found by experi- 
ence that the maximum obliquity should not in general exceed 30°, giving 
a mean of 15°. When more than one pair of teeth are in contact, a high 
maximum is less objectionable. 

As the number of teeth in a wheel decreases, they necessarily become 
longer to secure the proper path of contact, and both the obliquity of 
action and the sliding increase. From the preceding considerations the 



LOW-NUMBERED PINIONS, CYCLOID AL SYSTEM. 



215 



practical rule is deduced that, for millwork and general machinery, no 
pinion of less than twelve teeth should be used if it is possible to avoid it. 

It often becomes necessary, however, to use wheels having less than 
twelve teeth, in light-running mechanism, such as clockwork. In such 
cases a greater obliquity may be admissible, and for light work the flank- 
describing circle may be made large. 

Let it be required to determine the possibility of using two equal 
pinions, having six teeth, with radial flanks, the arcs of approach and 
recess each equal to one-half the pitch, and to find the maximum angle 
of obliquity. Fig. 298 is the diagram for two such gears. The path of 




Fig. 298. 

contact is to begin at a, the arcs ab, cb, and db each being equal to one- 
half the pitch; then the face of the pinion B must be long enough to be 
in contact with the flank of A at a. Drawing the line aef from a to the 
centre of B, we find that the distanced is less than one-half the thickness 
of the tooth, and that the approach is possible. Since the pinions are 
alike, the recess is also possible. The maximum angle of obliquity in 
approach is the angle 0, and this may be found in degrees as follows. 
The arc be on the pitch circle A subtends an angle bgc equal to one-half 
the pitch angle, the arc of approach being equal to one-half the pitch; 
in this case the angle bgc is 30°. The arc ab on the describing circle C 
is equal to be and therefore subtends an angle bha, which is to the angle 
bgc inversely as the radii of the respective circles. In this case these radii 



216 



CONSTRUCTION OF GEAR-TEETH. 



are as 2 to 1, making the angle bha equal to 60°. (It is important to 
notice that the line gc does not pass through the point a excepting in the 
single case of a radial flank gear.) The angle 6 between the tangent and 
the chord ab will always be one-half the angle ahb subtended by the arc 
ab. This gives the angle of obliquity 30°. Therefore we find that two 
pinions with six teeth and radial flanks will work with arcs of approach 
and recess each equal to one-half the pitch and with a maximum angle 
of obliquity of 30°. By allowing a greater angle of obliquity the teeth 
may be made a little longer and so give an arc of action greater than the 
pitch, which should be the case in practice. 

Two pinions with five teeth each will work with describing circles 
having diameters three-fifths the diameter of the pitch circles, and arcs 
of approach and recess each equal to one-half the pitch, as shown by 
Fig. 299, the path of contact beginning at a, the arcs ab, cb, and db each 




Fig. 299. 



being equal to one-half the pitch. The action is possible, since de is 
less than one-half the thickness of the tooth. The maximum angle of 
obliquity is 30°, the angle bgc being 36° and bha being | of 36°, or 60°. 
Two pinions with four teeth each will just barely work with describ- 
ing circles having diameters five-eighths the diameter of the pitch circle, 
and with no backlash, the arcs of approach and recess each being one- 



LOW-NUMBERED PINIONS, CYCLOIDAL SYSTEM. 



217 



half the pitch. Fig. 300 shows the diagram for this case, and the teeth 
are apparently pointed, which would be the case if de were just one-half 
the thickness of the tooth. To determine the possibility of the action 
the angle dfe may be calculated. It should not be greater than 22|° 
to allow the desired arc of approach. It will be found to be 22° 27' 19", 




Fig. 300. 



so that the action is just possible. The maximum angle of obliquity 
6 will be found to be 36°. 

A pinion with four teeth will work with a pinion having four teeth 
or any higher number, if the arc of action is not required to be 
greater than the pitch, the maximum angle of obliquity not exceeding 
36°. 

The requirements may be very different from the above in every 
respect; an arc of action greater than the pitch would usually be 
required; it might be desired to have the arc of recess greater than 
the arc of approach; it might not be admissible to have so great an 
angle of obliquity or to have the teeth cut under so far as a describing 



218 



CONSTRUCTION OF GEAR-TEETH. 



circle five-eighths the pitch circle would require. The results would 
of course vary with the conditions imposed. 

1 88. Arbitrary Proportions. — The teeth of gear-wheels may be 
designed by the preceding methods so as to fulfil any proposed conditions 
of approaching and receding action. In the majority of cases the exact 
lengths of the approaching and receding action are not important pro- 
vided they are long enough. It is a very common practice to make the 
whole length of a tooth a certain fraction of the pitch; the part which 
projects outside of the pitch circle being made a little less than that 
within to allow the proper clearance. 

None of the arbitrary proportions can be considered absolute, as 
the proper amounts of clearance and backlash depend on the precision 
with which the tooth outlines are laid out to begin with, and then on the 
accuracy with which the teeth are made to conform to these outlines. 

In the best cut gears manufactured to-day the teeth barely clear 
each other when the fronts are in contact, and in any case the allowance 
made should depend on the accuracy of workmanship. In cast gears 
more clearance is necessary to allow for irregular shrinkage and 
rapping, or for slight derangements of the mould. 

The following table gives some of the proportions in common use, 
P representing the circular pitch. 



Length 

Clearance 

Working depth 
Addendum. . . . 
Thickness. . . . 

Space 

Backlash 



1 




V*. 


p 


7m 


p 


7 ir 


p 


7 ir 


p 


V,i 


p 


V,i 


p 


7n 


p 



0.67 P 
0.07 P 
0.60 P 
0.30 P 
0.475P 
. 525 P 
0.05 P 



3 




V, 


p 


/l. 


p 


"/h 


p 


"/«, 


P 


Vi, 


p 


7„ 


p 


Vl5 


p 



. 750 P 

o.oeop+o.04' 

0.690 P-0.04' 
0.345 P-0.02 / 
0.470P-0.02 / 
0.530P + 0.02 / 
0.060P + 0.04' 



In the first three systems the percentage of backlash is constant, 
the backlash thus increasing with the pitch. It is better, however, to 
decrease the percentage of backlash as the pitch grows larger; for the 
larger the pitch, the smaller will be the proportion borne to it by any 
unavoidable error in workmanship. The last system (4) of Fairbairn 
and Rankine is based on this view of the proper proportions of back- 
lash; the amount of backlash given is, however, much larger than that 
now generally allowed. 

Teeth proportioned by any of the four systems will in general be of 
good shape, and answer their purpose. Should the wheel have less than 
twelve teeth, or should the exact amount of approaching or receding 
action be of importance, no arbitrary system is to be used, but the 
proper dimensions should be determined by the methods as explained. 



INVOLUTE SYSTEM. 219 

The backlash and clearance in any case should always be as small as 
the quality of workmanship will permit. 

Since theoretically the teeth can be in contact on both sides at once, 
the backlash has been disregarded in the diagrams. Should it be intro- 
duced, the thickness of the tooth, instead of being one-half the pitch, 
would be one-half the pitch minus one-half the backlash. 

In using the diametral pitch the working depth of the tooth is usually 
taken as twice the diametral pitch; the clearance is then commonly 
taken as one-eighth the diametral pitch, thus making the length of 
the tooth 2\ times the diametral pitch. This system is used by the 
Pratt & Whitney Co. In an 8-pitch wheel the working depth is 
| //= =i", the addendum is J", and the clearance is iXi= ^V'- The 
diameter of the " gear-blank/' or the addendum circle, could easily 
be found by adding two to the number of teeth on the wheel and then 
multiplying by the diametral pitch. Thus the diameter for a 60-tooth, 
8-pitch wheel-blank would be 

60+2 

189. Conditions for a Uniform Velcity Ratio. — It is to be observed 
that the conditions for a uniform velocity ratio would have been satisfied 
had the acting tooth curves been traced by a marking-point not in the 
circumference of the describing circle. And it is also to be observed 
that the tracing-point need not be carried by a circle ; any other describ- 
ing curve that would roll on the pitch circles would fulfil the above 
conditions just as well, the resulting curves having a common point 
of tangency and a common normal at that point which passes through 
the pitch point. Hence we may in general say that any proper tooth 
outlines must be such as can be simultaneously traced upon the planes 
of rotation of the two wheels, while in action, by a marking-point which 
is carried by a describing curve moving in rolling contact with both 
pitch circles. And conversely, for any set of proper tooth curves there 
is a corresponding describing curve. 

190. Spur Gearing, Involute System. — Generation of the Tooth 
Outline. — Let o 1 and 0, be the centres of the two wheels A and B, Fig. 
301, whose pitch circles are in contact at a. The angle of obliquity is 
constant in a pair of involute wheels, which means that the path of con- 
tact will lie on a straight line, which is called the line of obliquity. The 
tooth curves are not rolled from the pitch circles, but from circles called 
base circles, derived from the pitch circles as follows: Draw the line of 
obliquity bac making the given angle of obliquity bae with the tangent 
dae. From the centres of the wheels draw the circles C and D tangent 
to the line of obliquity at the points b and c respectively; these circles 
are the base circles. Draw the lines o t b and o 2 c from the respective 



220 



CONSTRUCTION OF GEAR-TEETH. 



centres to the tangent points of the base circles and the line of obliquity. 
Then 

o x a : o 2 a = o 1 b : o 2 c; (60) 

or , in a pair of involute wheels the radii of the base circles are directly 
proportional to the radii of the pitch circles. If the teeth can be rolled 
from the base circles in such a way as to give a constant a. v. ratio in- 
versely proportional to their radii, then the desired result of a constant 




Fig. 301. 

a.v. ratio inversely proportional to the radii of the pitch circles will be 
obtained. 

Imagine the base circles to be connected by inextensible bands be 
and fg similar to a crossed belt connecting a pair of pulleys, and assume 
that no slipping of the imaginary band occurs as the base circles are 
turned; also let the line be carry a marking -point. The curves which 
this marking-point would trace on the planes of the respective base 
circles would be suitable for tooth curves. Thus, if we assume the 
marking-point to start at b and move through a distance be, the point 
b on the base circle C would have moved to /, where bf is equal to 6c, 
and the marking-point will have traced the involute fc of the base circle 
C. At the same time a point g on the base circle D, eg being equal to 
cb, will have moved to c, and the marking-point b will have traced the 



PATH OF CONTACT. 221 

involute he of the base circle D. These two involutes will be suitable 
for tooth curves, for they will evidently give by their sliding action the 
same motion to the base circles as the base circles had while the curves 
were being generated, which was a linear motion of the circle C equal 
to the linear motion of D, and hence the a. v. of C : a. v. of D as o 2 c : ofi, 
and from equation (60) this a.v. ratio is the same as that of the pitch 
circles A and B in rolling contact. Therefore these involutes by their 
sliding action will give to the wheels an a.v. ratio the same as the pitch 
circles would give by their rolling contact. 

It will also be seen in Fig. 301 that the fundamental law of gearing 
is fulfilled, that is, the normal to the tooth curves at any point of con- 
tact passes through the pitch point, this common normal being always 
the line of obliquity. 

The involutes thus found form both the face and flank of the tooth; 
and although the face may be longer even until the teeth become pointed, 
the acting flank (that is, the part of the flank which can be in proper 
contact with the face of the other wheel) cannot pass inside the base 
circle, the additional part of the flank necessary to carry the tooth 
down to the root circle not being part of the involute. 

Knowing the pitch and the backlash, both of which are laid off on 
the pitch circles, the teeth may be drawn in as shown in Fig. 301, the 
addendum being limited, however, as will be seen in the following para- 
graph. 

191. Path of Contact. — Relation between the Path of Contact and 
the Arc of Action. — Limit of the Addendum. — As was seen in the pre- 
vious paragraph, the path of contact is on the line of' obliquity. It is 
limited, as in all systems, by the respective addendum circles, and the 
addendum would be a maximum when the teeth are pointed. In the in- 
volute system, however, the addendum circles are limited. Fig. 302 
shows the acting side of a pair of teeth as they appear when in contact 
at the point b, where the base circle C is tangent to the line of obliquity ; 
at the pitch point a ; and at the point c, where the base circle D is tangent 
to the line of obliquity. If the face ce of the wheel A is made longer 
(which it could be if the tooth were not yet pointed) , no apparent conflict 
would occur; -but if the wheels were turned further,. so that the point of 
contact would tend to be beytfnd c, the additional involute face has no 
longer an involute with which to be in gear, and the radial flank exten- 
sion, as drawn in Fig. 302, would not be conjugate to the additional invo- 
lute face. Conflict would then occur, for the curve conjugate to the 
involute would lie within the radial flank. This conjugate of the involute 
could be used, but one of the chief advantages of the involute system, 
which will be noticed later, would thereby be destroyed. 

Therefore, in Fig. 302, the contact cannot begin sooner than at the 
point b with A driving R.H., and cannot go beyond c; or, in a pair of 



222 



CONSTRUCTION OF GEAR-TEETH. 



external involute wheels the addendum is limited in each wheel by the tan- 
gent point of the line of obliquity and the base circle of the other wheel. 

To insure perfect action the arc of action must be at least equal to 
the pitch. In the involute system the path of contact is not equal to 
the arc of action, but is equal to the arc through which the base circles 
move while a pair of teeth are in contact, while the arc of action is the 
arc through which the pitch circles move in the same time. Since the 
pitch circles and base circles turn together, the arcs moved through 




Fig. 302. 

would subtend equal angles, and are proportional to the respective radii ; 
therefore 

path of contact _ arc moved through by base circle 
arc moved through by pitch circle 



arc of action 

_ radius of base circle 
radius of pitch circle 



cosine of angle of obliquity. 



(61) 



The last term of the above equation is obtained by noticing that the 
radius of the base circle from b and of the pitch circle from a are respect- 
ively perpendicular to the line of obliquity and to the tangent. 

The above relation is indicated in Fig. 302 for the arc of recess. The 
two teeth in contact at a move through the arc of recess ae when they 
will be in contact at c; the path of contact is ac equal to the arc fg, 



NORMAL PITCH. 223 

which the teeth move through on the base circle. Noticing that the 
arcs ae and fg must subtend equal angles at the centre of the wheel A, 
we have 

ac fg radius of base circle 



ae ae radius of pitch circle 



= cos cah. 



Thus for involute gearing the following law holds good: The path of con- 
tact is to the arc of action as the cosine of the angle of obliquity. 

By this relation we have a simple method for determining the beginning 
or end of the path of contact for any desired arc of action. In Fig. 302 
let it be required to find the end of the path of contact for a given arc 
of recess. Lay off from the pitch point a on the tangent a distance ah 
equal to the arc of recess; draw from h & line he perpendicular to the 
line of obliquity; the point c is the end of the path of contact, for, by 
construction, 

ac 7 path of contact in recess 

= cos cah = 



ah arc of recess 

192. Normal Pitch. — The normal pitch in an involute wheel is the 
distance from one tooth to the corresponding side of the next tooth 
measured on a normal to the curves, and from the method of generating, 
the curves this distance is a constant, and is equal to the distance between 
the corresponding sides of two teeth, measured on the base circle. 

In a pair of involute wheels the path of contact cannot be less than 
the normal pitch, the corresponding arc of action being the circular pitch, 
for it will be evident, from the discussion in the preceding paragraph, 
that 

normal pitch 



circular pitch 



= cosine of angle of obliquity. . . . (62) 



193. To determine the possibility of any desired action in a pair 
of involute wheels. — The solution is similar to that in the cycloidal 
system, the essential difference being due to the fact that the path 
of contact is not equal to the arc of action. In Fig. 303 let it be 
required to determine if the arc of recess can be equal to } of the 
pitch. Lay off from a on the tangent the distance a6 = J of the 
pitch. Draw be perpendicular to the line of obliquity; c will be 
the end of the path of contact for the given arc of recess. If 
the point c came beyond d, the tangent point of the line of ob- 
liquity and the base circle, the action would be impossible since no 
contact can occur beyond d. But if, as in Fig. 303, the point c 
comes between a and d, it is necessary to determine if the face of the 
tooth on A can reach to c. Lay off on the pitch circle A the arc 
ae = ab = \ of the pitch; the face of the tooth on A will then pass 
through c and e. Draw the line co from c to the centre of A, and 



224: 



CONSTRUCTION OF GEAR-TEETH. 



note the point / where it cuts the pitch circle A. If ef is less than 

one-half the thickness of the tooth, 
the action can go as far as c and 
the teeth will not be pointed. In 
the figure, assuming no backlash, the 
thickness of the tooth would be eg, 
and ef is less than \eg; therefore the 
action is possible, as is shown by 
the two teeth drawn in contact 
at c. 

194. Involute Pinion and Rack. 
— Fig. 304 shows a pinion driving 
a rack. The path of contact cannot 
begin before the point a, but the 
recess is not ♦ limited excepting by 
the addendum of the pinion, since 
the base line of the rack is tangent 
to the line of obliquity at infinity. 
For the same reason it will be evident 
that the sides of the teeth of the 
rack will be straight lines perpendicu- 
lar to the line of obliquity. In the 
figure the addendum on the rack is 

made as much as the pinion will allow, that is, so that the path of 





Fig. 304. 

contact will begin at a. The addendum of the pinion will give the end 
of the path of contact at b. 



INVOLUTE PINION AND ANNULAR WHEEL. 



225 




In Fig. 305, the diagram for a pinion and a rack, let it be required 
to determine if the path of contact can begin at a and go as far as b; 
to be solved without using the 
tooth curves. The solution is 
similar to that in § 193. For 
the contact to begin at a the 
face of the rack must reach to 
a. Draw the line ac perpen- 
dicular to the line of obliquity, 
giving cd as the arc of ap- 
proach; draw ae parallel to 
the line of centres, and if ce 
is less than one-half the thick- 
ness of the rack tooth, the 
approaching action is possible 
without pointed teeth. Simi- 
larly for the recess, draw the Fig. 305. 
line bf perpendicular to the line of obliquity, giving df equal to the arc 
of recess; make the arc dg on the pinion's pitch circle equal to df, then 
the face of the pinion's tooth will pass through b and g; draw the line 
bh to the centre of the pinion, and note the point h where it crosses 
the pinion's pitch circle. If -gh is less than one-half the thickness of the 
tooth, the recess is possible without pointed teeth. 

195. Involute Pinion and Annular Wheel. — Fig. 306 shows an involute 
pinion driving an annular wheel. This case is very similar to a pinion 
and rack. The addendum of the annular is limited by the' tangent point 
a of the pinion's base circle and the line of obliquity, while the adden- 
dum of the pinion is unlimited except by the teeth becoming pointed. 
The base circle of the annular lies inside the annular, so that its point 
of tangency with the line of obliquity is at b. If we take some point 
on the line of obliquity, as e, and roll the tooth curves as they would 
appear in contact at that point, the teeth of the annular will be found 
to be concave, and the addendum of the annular will seem to be limited 
by the base circle of the annular where the curves end. But if these 
two teeth are moved back until they are in contact at a, it will be evident 
that the annular's tooth curve cannot be extended beyond a without 
interfering with the radial extension of the flank of the pinion. There- 
fore the addendum of the annular is limited by the point of tangency of 
the base circle of the pinion and the line of obliquity. 

196. Possibility of Separating two Involute Wheels. — One of the most 
important features of involute gearing is the fact that two such wheels 
may be separated, within limits, without destroying the accuracy of the 
a.v. ratio. In this way the backlash may be adjusted, since the original 
pitch circles need not be in contact. To show that this is so, Fig. 301 



226 



CONSTRUCTION OF GEAR-TEETH. 



may be redrawn using the same pitch circles and base circles, but separat- 
ing them slightly, keeping the teeth in contact, as has been done in Fig. 
307. Connect the base circles by the tangent be. If now the line be 
carries a marking-point, it will evidently trace the involutes of the two 
base circles, as de and he, and these curves must be the same as the tooth 




Fig. 306. 

curves in Fig. 301. In Fig. 307 these curves de and he will give an a. v. 
ratio to the base circles inversely as their radii, but the radii of these base 
circles are directly as the radii of the original pitch circles (Fig. 301); 
hence in Fig. 307 the tooth curves de and he would give an a. v. ratio to 
the two wheels inversely as the radii of the original pitch circles, although 
these circles do not touch. The path of contact is now from k to e, 
which is considerably shorter than in Fig. 301 ; it is, however, greater than 
the normal pitch, so that the action is still sufficient. The limit of the 
separation will be when the path of contact is just equal to the normal 
pitch. The angle of obliquity is bam, which is greater than in Fig. 301. 
The backlash has also increased. 

Theoretically the wheels have new pitch circles in contact at a, and 
a new angle of obliquity, also a greater circular pitch with a certain 
amount of backlash; and if we had started with these latter data, we 
should have obtained exactly the same wheels as in Fig. 301, only slightly 
separated. It will be seen that the radii of the new pitch circles are to 
each other as the radii of the respective base circles, and consequently as 



POSSIBILITY OF SEPARATING TWO INVOLUTE WHEELS. 227 

the respective original pitch circles. It will also be seen that the line of 
obliquity, which is the common normal to the tooth curves, passes 
through the new pitch point o so that the fundamental law of gearing 
is still fulfilled. 

By the application of the preceding principles two or more wheels 
of different numbers of teeth, turning about one axis, can be made to 
gear correctly with one wheel or one rack; or two or more parallel racks 




Fig. 307. 



with different obliquities of action may be made to gear correctly with 
one wheel, the normal pitches in each case being the same. Thus dif- 
ferential movements may be obtained which are not possible with teeth 
of any other form. 

The principal objection to the use of involute teeth for large gears 
is the great obliquity of action and the large number of teeth in the 
smallest wheel. 

On the other hand, for smooth action, especially for light work, it 
may be an advantage to have this constant obliquity of action, as the 
side pressure will tend always to keep the axes at the greatest possible 
distance from each other, thus preventing jarring in case there be any 
looseness in the bearings. 



228 



CONSTRUCTION OF GEAR-TEETH. 



107. Interchangeable Involute Gears. — Since the tooth curve in an 
involute wheel depends only on the base circle, or in other words on the 
diameter of the pitch circle and the angle of obliquity, wheels may be 
designed separately, keeping the normal pitch the same in each wheel 
of a set, the addendum and clearance being so taken that no interference 
will occur with the smallest wheel desired in the set. 

Let it be required to design an involute wheel having 36 teeth and 
6-P., with an obliquity of 15°, the smallest wheel of the set to allow arcs 

of approach and recess each 
equal to the pitch. In Fig. 
308, A is the pitch circle, 
bac the line of obliquity, 
and B the base circle of the 
36-tooth wheel. The pitch 
is approximately 0".52, and 
is laid off on the tangent 
on each side of a so that 
ad=ae = 0".52. The de- 

sired - path of contact will 
then be found to be bac, 
where ba = ac = the normal 
pitch. The addendum for 
the 36-tooth wheel must 
pass through b, and should 
not be greater, for the small- 
est wheel of the set would 
theoretically have its base 
circle tangent to the line of 
obliquity at b, just allowing 
the path of contact to go 
to b. This would give the 
centre of the smallest 
pinion of the set at o 2 , 
provided, however, that the 
distance ao 2 can be used 
with the given pitch. In 
this example the wheels are 
6-P., so the radius of the 
pitch circle must be divisible 
into twelfths of inches. 
This will generally throw 
the actual centre a little 
beyond the centre found in the diagram. The addendum circle of the 
smallest wheel will pass through c; this will give the root circle C for 




INTERCHANGEABLE INVOLUTE GEARS. 229 

the 36-tooth pinion, the clearance being known. We have thus the com- 
plete diagram for the 36-tooth wheel and may proceed to roll the teeth. 

Since the path of contact is not intended to go beyond c,the involutes 
need not be extended to the base circle, and instead of using radial 
extensions from the involutes to the root circle, straight lines tangent 
to the involutes at the points where a circle F, drawn through the point 
c, cuts them, may be used. If a tooth were just ready to begin action 
at c, as shown in Fig. 308, the line of obliquity would, at that position 
of the tooth, be normal to the acting involute ; therefore if we draw the 
line cf from c perpendicular to the line of obliquity, it will give the direc- 
tion of the desired flank extension at that position. If a circle is now 
drawn with o x as a centre and tangent to the line cf, the flank extensions 
for all the teeth will be tangent to this circle, as shown in the figure for 
one tooth. This will result in a stronger form of tooth than that obtained 
with a radial extension. 

When interchangeable gears are constructed with involute teeth, the 
addendum is made the same for all wheels, and is usually taken, as in 
the epicycloidal system, equal to the diametral pitch; but -when the 
addendum is thus arbitrarily chosen, the teeth of the larger wheels, 
particularly of the rack, would be liable to conflict with the flank exten- 
sions of the smaller wheels; to avoid such interference the ends of the 
teeth must be rounded off. 

198. Arbitrary Proportions for Involute Gears. — Rankine's rule to 
make the entire length of the tooth 0.75P, or that of Willis to make 
it 0.7P, will give good results. Experience has shown that for ordinary 
purposes the obliquity should not exceed from 15° to 17°. When it 
is no greater than this it is safe to say that these teeth are equal to the 
cycloidal or any others, even for heavy work; in addition to other advan- 
tages that have been mentioned, it is to be observed that the form of 
the tooth is a strong one. 

199. Pin Gearing. — In this form of gearing the teeth of one wheel 
consist of cylindrical pins, and those of the other of surfaces parallel to 
cycloidal surfaces, from which they are derived. 

In Fig. 309 let o t and o 2 be the centres of the pitch circles whose 
circumferences are divided into equal parts, as ce and eg. Now if we 
suppose the wheels to turn on 'their axes, and to be in rolling contact 
at c, the point e of the wheel o t will trace the epicycloid gp on the plane 
of the wheel o 2 , and merely a point e upon the plane of o v Let cf be 
a curve similar to ge and imagine a pin of no sensible diameter — a rigid 
material line — to be fixed at c in the upper wheel. Then if the lower 
one turn to the right, it will drive the pin before it with a constant velocity 
ratio, the action ending at e if the driving curve be terminated at / as 
shown. 



230 



CONSTRUCTION OF GEAR-TEETH. 



If the pins be made of a sensible diameter, the outlines of the teeth 
upon the other wheel are curves parallel to the original epicycloids, as 
shown in Fig. 310. The diameter of the pins is usually made about 
equal to the thickness of the tooth, the radius being, therefore, about J 
the pitch arc. This rule is, however, not imperative, as the pins are 
often made considerably smaller. 





Clearance for the pin is provided by forming the root of the tooth 
with a semicircle of a radius equal to that of the pin, the centre being 
inside of the pitch circle an amount equal to the clearance required. 

The pins are ordinarily supported at each end, two discs being fixed 
upon the shaft for the purpose, as shown in Fig. 274, thus making what is 
called a lantern wheel or pinion. 

Mode of Action. — In wheel work of this kind the action is almost 
wholly confined to one side of the line of centres. In the elementary 
form (Fig. 309) the action is wholly on one side, and receding, since 
it cannot begin until the pin reaches c (if o 2 drives), and ceases at e; 
if o x is considered the driver, action begins at e, ends at c, and is wholly 
approaching. As approaching action is injurious, pin gearing is not 
adapted for use where the same wheel has both to drive and to 
follow; the pins are therefore always given to the follower, and the 
teeth to the driver. 

When the pin has a sensible diameter, the tooth is shortened and 
its thickness is decreased ; the line of action is also shortened at e, Fig. 310, 
and, instead of beginning at c, will begin at a point where the normal 
to the tooth curve, through the centre of the pin, first comes in con- 
tact with the derived curve mf. This normal's end will not fall at c, 
but at a point on the arc ce beyond, on account of a property of the 
curve parallel to an epicycloid. The parallel to the epicycloid is shown 



PIN GEARING. 



231 




Fig. 311 



in Fig. 311, cp being the given epicycloid. The curve may be found by 
drawing a series of arcs ss with a radius equal 
to the normal distance between the curves, and 
with the centres on cp. The parallel curve first 
passes below the pitch curve cm and then rises, 
after forming a cusp, and cuts away the first 
part drawn : this is more clearly shown somewhat 
exaggerated at mno. Hence the part which 
would act on the tooth when its centre is at c is 

cut away, and, for the same epicycloid, the greater the diameter of the 
pin the more this cutting away. In Fig. 310 the pin e is just quitting 
contact with the tooth at i while c is at the pitch point, and, according to 
the above property of the parallel to the epicycloid, is not yet in contact 
with the tooth m. Strictly speaking, then, the case shown is not a 
possible one, as the tooth should not cease contact at i until m begins its 
action. The above error is practically so small that it has been disre- 
garded, especially for rough work. 

The following method may be used in determining a limiting case 
in pin gearing: 

If we assume the pitch arc = eg (Fig. 312), the greatest possible 
height of tooth is determined by the intersection of the front and back 

of the tooth at p; and if this height is 
taken, action will begin at c and end 
at h, the point in the upper pitch circle 
through which p passes. Now if p falls 
upon the pitch circle ceh, we should have 
a limiting case for a pin of no sensible 
diameter. If the pin has a sensible 
diameter and the pitch arc cg = ce is 
assigned, bisect eg with the line o 2 p and 
draw ce intersecting o 2 p in k; assume a 
radius for the pin less than ek and draw 
the derived curve to cut o 2 p in j } which 
will be the point of the tooth. Through 
j draw a normal to the epicycloid, cut- 
ting it at s; through s describe an arc 
about o 2 cutting the upper pitch circle at 
t , the position of the centre of the pin at 
the end of its action. Draw the out- 
line mf of the next working tooth, find 
the point m at the cusp of the carve 
parallel to the epicycloid, and draw the normal mn; m is the lowest 
possible working point of the tooth. Through n describe an arc about o 2 
cutting the original path of contact in r, which is the point that n must 




Fig. 312. 



232 



CONSTRUCTION OF GEAR-TEETH. 



reach before the tooth will be in contact with the pin, or is the point that 
n must reach before the common normal to the pin and tooth curve 
passes through the pitch point. 

Now action begins when the axis of the pin is at r and ends at t; if 
rt = ce, we have an exact limiting case and the assumed radius of the pin 
is a maximum; if rt<ce, the radius is too great; but if rt>ce, the case is 
practical. To get the exact limit a process of trial and error should be 
resorted to. When the pin is a point the methods used in cycloidal 
gearing may be applied; the correction for a pin of sensible diameter 
can then be made by applying the method of Fig. 312. 

Wheel and Rack. — As the pins are always given to the follower, 
we have two cases. 

1° Rack drives, giving the pin-wheel and rack, Fig. 313. Here the 





Fig. 313. 



Fig. 314. 



original tooth is bounded by cycloids generated by the pitch circle of the 
w T heel. 

2° Wheel drives, giving the pin-rack and wheel. Here (Fig. 314) the 
original tooth outline is the involute of the wheel's pitch circle. 





Fig. 315. 



Fig. 316. 



Inside Pin Gearing. — Here also there are two cases. 

1° Pinion drives (Fig. 315). The original tooth outlines will be 



DOUBLE-POINT GEARING. 



233 



internal epicycloids generated by rolling the pitch circle of the annular 
wheel on the pinion's pitch circle. 

2° Annular wheel drives (Fig. 316). Here the original tooth outline is 
the hypocycloid traced by rolling the pinion's pitch circle in the wheel's 
circle. 

Path of Contact. — In the elementary form of tooth (Fig. 309) the 
path of contact is on the circumference of the pitch circle of the follower 
o v as ce. When a pin is used its centre always lies in this circumference, 
and its point of contact may be found by laying off a distance ei equal 
to the radius of the pin (Fig. 317) on the common normal. Drawing 





Fig. 318. 



a number of these common normals, all of which must pass through the 
pitch point c, and laying off the radius of the pin ei on each, we have 
the path of contact ci known as the limagon. 

200. Double-point Gearing. — This form of gearing, shown in Fig. 318 r 
gives very smooth action where not much force is to be transmitted. 
The pitch circles are here taken as the describing circles ; the face eg of 
the pinion o^ is generated by rolling the pitch circle o 2 on that of o l} and 
the face cf is generated by rolling the pitch circle o 1 on o 2 . If o 1 is con- 
sidered the driver, action begins at d, the point c of o x sliding down the 
face cf while c travels from d to c. In the receding action the point c of 
the tooth of o 2 is acted on by the face eg while c moves from c to e. The 
spaces must be so made as to clear the teeth. This combination reduces 
friction to a minimum and .gives the obliquity of action less than in any 
case except pin gearing, but the teeth are much undercut and weakened 
by the clearing curves, and if much force is to be transmitted the line of 
contact will soon be worn away. 

Shrouded Wheels. — When the teeth of a wheel, as o lf Fig. 318, are 
undercut and weak they are sometimes united at their ends* by annular 
rings cast with the wheel, and the wheel is then said to be shrouded. 



234 CONSTRUCTION OF GEAR-TEETH. 

This shrouding strengthens the teeth and is usually applied to the pinion, 
where the wear is greater; it may extend the whole depth of the teeth 
of the pinion, or both pinion and wheel may be shrouded to half the 
length of the teeth. The latter arrangement is seldom adopted on 
account of the difficulty of casting the wheels. 

201. Sang's Theory. — A conjugate curve has already been defined, 
§ 177. If in Fig. 278 we assume any tooth outline, as aa v we may con- 
struct a series of wheels, assuming different pitch circles, that will work 
with the wheel having the assumed tooth outline. Then any one of the 
series may be taken and a second series of wheels made from its tooth 
outline. From the method of generating the tooth outlines it follows 
that any wheel of the first series will gear with anyone of the second. 
Now if the conjugate tooth outlines on any two equal wheels be made the 
same, the two series will be identical and the wheels will all be inter- 
changeable. If the tooth and space are taken equal to each other, two 
racks formed from the above equal wheels will exactly fit into each other, 
the tooth of one filling the space of the other and being at the same 
time equal to that of the other. 

Hence we may assume the outline of a rack tooth such that it is made 
up of four equal lines in alternate reversion (thus giving a symmetrical 
tooth about its centre line) and from it derive an interchangeable set of 
wheels. If the rack be made up of equal cycloidal arcs, the cycloidal 
system with a constant describing circle will be obtained; if it be 
bounded by oblique straight lines, the involute system will be repro- 
duced. 

The rack may be arbitrarily assumed, provided it fulfils the above 
conditions, and different series of wheels may be derived. This method 
of constructing the tooth outline, which is practically the reversal of the 
ordinary way, is due to Professor Sang. 

Should we now shape a cutter to the exact outline of a rack tooth 
and then give to it a reciprocating motion, parallel always to the axis of 
a gear-wheel blank, which is made to move in reference to the rack 
tooth just the same as it would if the pitch surface of the rack and that 
of the wheel were in rolling contact with each other, the rack tooth would 
shape the space of the wheel; it being understood that the wheel is not 
moved Avhile the rack tooth is cutting, and that it is only rolled fast 
enough to give a light chip to cut each time the tooth moves in the cutting 
direction. One space having been cut, the blank is turned through the 
pitch angle and the operation is repeated until the wheel is formed. 
This method has been applied in a bevel-gear-cutting engine. 

202. Unsymmetrical Teeth. — In all the figures hitherto given the 
teeth are symmetrical, so that they will act equally well whether the 
wheels are turned one way or the other. In cases where the action 



TWISTED GEARING. 



235 




is always one way it may be advantageous to make the teeth otherwise, 

as shown in Fig. 319. Here the lower wheel, 

o 2 , is the driver, and the acting outlines of 

both wheels are of the cycloidal form; the 

describing circles p x and p 2 have been taken 

large to reduce the obliquity to a minimum. 

If the other sides of the teeth were made the 

same, we should have a weak tooth at the 

root. To avoid this the backs of the teeth 

may be made involutes of considerable 

obliquity, the radii of the base circles being 

o x a and o 2 b. It can be seen that this gives a 

very strong form of tooth. F IG . 319. 

203. Twisted Gearing. — Hooke's Stepped Wheels. — If a pair of spur- 
wheels are cut transversely into a number of plates, and each plate is 
rotated through an angle, equal to the pitch angle divided by the num- 
ber of plates, ahead of the adjacent plate, as shown in Fig. 320, we shall 

have a pair of stepped wheels, first intro- 
duced by Dr. Hooke. By this device we 
obtain the effect of increasing the number 
of teeth without diminishing their strength ; 
the number of contact-points is also in- 
creased, and the interval between their 
times of crossing the line of centres, where 
the action is best, is correspondingly di- 
minished. The upper figure shows a sec- 
tion on the pitch line A A. The action for 
each pair of plates is the same as that for spur-wheels having the same 
outlines. In practice there is a limit to the reduction in the thickness of 
the plates, depending on the material of the teeth and the pressure to be 
transmitted, since too thin plates would abrade. The number of divi- 
sions is not often taken more than two or three, and the teeth are thus 
quite broad. These wheels give a very smooth and quiet action. 

Hooke's Twisted Gearing. — If the number of plates be taken infinite, 
the effect is the same as that explained in § 174, 4°. The twisting being 
uniform, the elements of the teeth become helices, all having the same 
pitch. The line of contact between two teeth will have a helical form, 
but will not be a true helix; the projection of this helix on a plane per- 
pendicular to the axis will be the ordinary path of contact. It can easily 
be seen that the common normal at any point of contact can in no case 
lie in the plane of rotation, but will make an angle with it. The line of 
action then can in general have three components: 1° A component 
producing rotation, perpendicular to the plane of the axes; 2° A com- 
ponent of side pressure, parallel to the line of centres; 3° A component of 




Fig. 320. 



236 



CONSTRUCTION OF GEAR-TEETH. 




end pressure parallel to the axes. When the point of contact is in the 
plane of the axes the second component is zero; advantage may betaken 
of this, as will be shown, so that there may be no sliding action between 
the teeth. The end pressure is neutralized as explained in § 174, 4°. 

Sliding Friction Eliminated. — In this case the angle of twist is at 
least equal to the pitch angle and should be a little more. In Fig. 321, 
which represents a transverse section of a pair of twisted wheels, sup- 
pose the original tooth outlines to have been 
those shown dotted. Then cut away the faces 
as shown by full lines having the new faces tan- 
gent to the old ones at the pitch point c; we 
now have lost proper contact except that at c 
for the section shown, but by twisting the 
wheels this contact can be made to travel along 
the common element of the pitch cylinders 
through c from one side of the wheel to the 
Fig. 321. other. A simple construction to use in this 

case is to make the flanks of the wheels radial and the faces semicircles 
tangent to the flanks. The action here is purely rolling and is very 
smooth and noiseless; but for heavy work it is best to use the common 
forms of teeth with sliding action, so that the pressure may be distrib- 
uted over a line instead of acting at a point. 

204. Approximate Forms of Teeth. — To secure perfect smoothness of 
action in toothed wheels, it is necessary that the tooth outlines should 
be accurately laid out, as explained in the preceding pages, and that the 
teeth when constructed should conform exactly with the outlines found. 
If the teeth are to be cut, the exact curves should be used, as when the 
cutter is once made it will cut the accurate shape -as well as any other. 
When, however, the teeth are to be cast, or for some other reason perfect 
accuracy is not required, the exact curves may be replaced by others 
which approximate to them more or less closely, but which are simpler 
to construct. This is possible as, the teeth being short, only a small 
part of the theoretical curve is used. In these approximations the pro- 
portions of the teeth are usually 
governed by one of the sets of 
arbitrary proportions given in 
§ 188. The two principal methods 
of approximation are: 1° by cir 
cular arcs, and 2° by curved tern 
plates. 

Approximation by Ci rcular 
Arcs. — Wi1}is M-ethod. — Let and 
o x (Fig. 322) be the centres of 
two pitch circles in contact at c. 




Fig. 322. 
Draw a line qcq x making an angle 6 



APPROXIMATE FORMS OF TEETH. 237 

with the line of centres, and through c draw the line ck perpendicular 
to qq t . On ck assume any point, as k, and through this point draw 
the lines ko and ko x q intersecting qq x at p and q respectively. These 
points may now be taken as limiting the length of the connecting-rod of 
a four-bar linkage opqo v the links op and o t q turning about o and o t 
respectively, k being the instantaneous axis of pq v For the a. v. ratio 
we have 

a.v. op _o x c 

a.v. o x q oc' 

which is the same as that for the rolling pitch circles. This angular 
velocity ratio is also momentarily constant, as ck is perpendicular to pq 
(§98, page 75) ; and for a slight angular movement of the links either way 
from their present position pq would still pass through c. If now through 
any point, as m, on pq we draw two circular arcs, as mn and mt, with p 
and q as centres respectively, they will do for tooth curves, since they 
will retain p and q at a distance = pm + mq=pq apart, thus replacing the 
link, and will also have, for a limited motion, their common normal at 
the point of contact passing through the pitch point c. In the figure mn 
might be considered the face of o, and mt the flank of o v Had the 
point m been taken outside of pq, both arcs would have been convex the 
same way. If o x be placed so that the angle ko x c is acute, as, for example, 
at o 2 , then q will fall at q x on the same side of c as p, and this will make 
the flank mt concave instead of convex. But if ko 1 becomes parallel 
to ps, then q will fall at an infinite distance from p, and mt will become 
perpendicular to ps. 

Application to Involute Teeth, where the outline of the tooth con- 
sists of a single arc. — In Fig. 322 let ck become infinite; then op and o x q 
will become perpendicular to pcq, and the points p and q will be found at 
r and s respectively. Let the arcs of the teeth be struck through c, and 
let ocr = 6, the radius of the wheel oc = R, and the required radius of tooth 
outline cr = D; then D = R cos d, which is independent of the wheel o 1} as 
well as of the pitch and number of teeth of o. If, then, the angle 6 and 
the pitch be made constant in a set of wheels, any two wheels of the set 
will work together. 

Assume 6 = 75° 30', a very convenient value; then 

D = R cos 75° 30' = RX 0.25038 = — , very nearly. 

To apply this approximation, let oc (Fig. 323) be the radius of the 
pitch circle of the proposed wheel. Draw cp, making an angle ocp = 
75° 30' with oc, and drop the perpendicular op upon cp, or, better, 
describe a semicircle on oc, and make the chord cp=\oc; then p will 
be the centre for the tooth outline acb drawn through c. The tooth 
may now be completed as shown. The centres of all the curves are 



238 



CONSTRUCTION OF GEAR-TEETH. 



found on the circumference of a circle of a radius op, and the lengths 

of the teeth should be kept 
within the limits mentioned in 
§191. 

For convenience a bevel 
template, as shown at T, may 
be made, the angle ocs being 
75° 30'. The edge cs can 
then be graduated J size ; now, 
knowing the radius of the 
wheel, the position of p may 
be found directly by adjusting 
the template as shown, and 

noting the point p at the radius reading on the scale. 

Application to Cycloidal Teeth where the side of the tooth consists 

of two circular arcs. Let o (Fig. 324) be the centre of the given wheel, 




Fig. 323. 




Fig. 324. 

o l that of a wheel with which it is to gear, and c the pitch point. Draw 
qcq x , making an angle 6 with oo v and through c draw the perpendicular 
kck v making kc = k x c and less than either oc or o x c. Draw ok and o x k, cutting 
qc at p and q t respectively. Lay off cm = ^ pitch on the pitch circle AA 1 on 
the side of c opposite p and q x ; then p is the centre and pm the radius of 
the face of o drawn through m, which face will work with a flank of o x with 
a centre q t and radius q x m. Lines from o 1 and o through k x locate the 
tooth centres p t and q; then laying off cw = ^ pitch on A, we have p x n, 
the radius of the faces of o XJ and qn, the radius of the flanks of o. Circles 
pf and qs drawn through p and q about o will locate the face and flank 
centres respectively for the wheel o, and circles through p 1 and q x about o x 



APPROXIMATE FORMS OF TEETH. 



239 



will locate the face and flank centres for the wheel o v If now the points 
k and k t remain fixed, changing the radius o x c will not affect p and q, 
the centres of the tooth curves for o ; hence any number of wheels may 
be designed, using different values for o x c, that will work with the wheel 
o. To find the limit of ck 1 for a given value of oc, we see from the figure 
that when o approaches c, cq increases, becoming infinite when ok x is 
parallel to cq, thus giving flanks perpendicular to cq through n. If 
o approach still nearer c, the flanks become convex {q then appearing 
above c), which would give an awkward tooth form. The greatest 
value given to ck t is then that which makes ok t parallel to cq, and the 
smallest wheel of the set will have straight flanks. If the radius of 
this smallest wheel is represented by R , and if D represents the distance 
pc, and d the distance qc, then, by assuming values for R and in a set 
of wheels, the corresponding values of D and d for different pitches and 
numbers of teeth may be calculated and arranged in tabular form. 
Professor Willis assumed 6 = 75°, and took twelve as the least number of 
teeth to be given to any wheel, the flanks of this wheel being radial. 

The Willis Odontograph consists of two thin strips T (Fig. 325) 
making an angle of 75° with each other; the edge nr corresponds to 
oc, Sindnq to cq (Fig. 324). The edge nq is graduated with equal divisions 
beginning at n and going both ways. The graduations to the right of n 
are for face centres, and to the left of n for flank centres; these gradua- 
tions are made to suit the tables calculated by the method suggested 
above. 

Fig. 325 illustrates the method of applying the instrument. The pitch 
and number of teeth being known, the 
radius of the pitch circle oc can be 
found ; make mn equal to the pitch arc 
and bisect it in c; find from the tables 
the values D = mp and d = nq, which 
locate the centres p and q respectively 
for the face ca and the flanks cb. 
The method of using the instrument 
can easily be seen from the figure. 

Wheels laid out with the odonto- 
graph resemble the cycloidal wheels 
with a constant describing circle, of 
a diameter one-half that for a twelve- 
toothed pinion. The outlines of the 
teeth show an angle at the pitch points of the teeth. 

The approximate radii of the face and flank curves for teeth, with 
the radii of their centre circles pf and qs, Fig. 324, may be calculated 
from the values of D and d ; tabulating these, we may get along without 
the odontograph. This has been done by Mr. George B. Grant, who 




Fig. 325. 



240 CONSTRUCTION OF GEAR-TEETH. 

has arranged a table which gives the radii of tooth curves and the radial 
distances between the pitch circle and the face, and the flank centre- 
circles. 

Mr. Grant has also arranged a table, known as " Grant's odonto- 
graph table," in which the approximate circular arcs are made to con- 
form more nearly to the theoretical shape than by the Willis method. 
The Willis arc lies wholly within the true curve, while the Grant arc inter- 
sects the tooth face in three points; viz., at the pitch line, at the addendum 
line, and at a point midway between. The above tables may be found 
in a A Handbook on the Teeth of Gears," by George B. Grant. 

Robinson's Template Odontograph. — This ingenious instrument, the 
invention of Prof. S. W. Robinson, gives the outline of the tooth direct, 
and may be used in the pattern-shop for laying out gear patterns. It 
was found that the curve, to satisfy the mathematical conditions in 
what precedes, 1° must be one of rapidly changing curvature, approxi- 
mating very closely to the epicycloid ; 2° it must be very nearly perpen- 
dicular to the pitch circle at the middle point of the tooth outline; and 
3° it must intersect the addendum circle at the same point as the epic}^- 
cloid; in short, it must coincide with the epicycloidal face. The curve 
most completely satisfying these conditions was found to be a logarithmic 
spiral. 

The odontograph consists of a thin brass plate fgh (Fig. 326), grad- 
uated on the edge gh, the figure showing the instrument about one-sixth 

k size. Accompanying the instru- 

y/^\ ment are tables varying according 

4f~~ s V to the kind of tooth desired. Fig. 

\ \. 326 shows the method of using the 

^r^^-^^X odontograph to lay out a wheel 

/ ^^v\ ) belonging to an interchangeable 

/ ^ )r^a series. The table is here arranged 

/ in four columns, giving: 1° Diam- 

' eter in inches ; 2° Number of teeth ; 

3° Face settings; 4° Flank set- 
tings. Let lc\ be the pitch circle, which is known when the pitch an 1 
number of teeth are given; assume c the middle point of a tooth, and 
lay off the arc ce = its half- thickness. Draw tangents ct and es to the 
pitch circle at c and e. Set the instrument in the position fgh, the 
proper division on the scale, found from the column of face settings, 
being brought to d while at the same time the curved edge fg is tangent 
to ct, and e is on the edge gh; now draw the face ea. To draw the 
flank the instrument is placed in the position f^Jh, the proper flank 
reading being at e, and the curve / 1 gr l being tangent to es. 

Professor Robinson's paper may be found in Van Nostrand's Maga- 
zine, Vol. 15. 



BEVEL GEARING. 



241 




Fig. 327. 



Prof. J. F. Klein has recently introduced a method of finding correct 
tooth outlines by means of tables specially prepared for the different 
systems of gearing. The method consists in giving, by table, the dis- 
tances of points of the tooth outline 
from each of two perpendicular refer- 
ence lines XX and YY (Fig. 327) drawn 
through some easily fixed point in the 
tooth outline. Each of the two sets of 
distances is expressed in simple decimal 
fractions of the pitch or diameter, and, 
for ease in tabulation, computation, and 
laying out, one of these sets is arranged 
in groups of equidifferent values. The 
computations from the tables are very 
simple, only short division or multipli- 
cation being necessary for determining 
ordinary outlines. After making the 
computations, the drawing of the out- 
line only involves the use of the square 
and the ability to lay off distances accurately. Tables are also given by 
means of which two reference lines, one on each side of a tooth, may be 
located in proper relation to each other, thus making it unnecessary to 
use compasses. This method is especially useful in laying out the out- 
lines of the teeth of large wheels w r here it would be inconvenient to roll 
up the curves in the ordinary way. 

205. Bevel Gearing. — In the discussion on the teeth of spur-wheels, 
the motions were considered as taking place in the plane of the paper, 
and we have thus dealt with lines instead of surfaces. But the pitch 
and describing curves, and also the tooth outlines, are but traces of 
surfaces acting in straight-line contact, and having their elements per- 
pendicular to the plane of the paper. In bevel gearing the pitch sur- 
faces, are cones, and the teeth are in contact along straight lines, but 
these lines are perpendicular to a spherical surface, and all of them pass 
through the centre of the sphere, which is at the point of intersection of 
the two pitch cones. 

As in spur gearing an element of the same rolling cylinder generates 
the faces of the teeth of the driver, and the flanks of the teeth of the 
follower, so in bevel gearing an element of the same rolling cone, having 
its apex at the point of intersection of the axes of the two pitch cones, 
generates the faces of the driver's and the flanks of the follower's teeth 
and vice versa. 

Let o-uc and sct-o (Fig. 328) be the pitch cones of a pair of bevel 
gears in contact on the line oc. Let o-ec be" a third cone in contact 
externally with sct-o, and internally with o-uc on the line oc. The above 



242 



CONSTRUCTION OF GEAR-TEETH. 



three cones, being in contact on oc, will have their axes in a plane pass- 
ing through oc. Suppose the bases 
of the cones to be circular portions 
of a spherical surface whose centre is 
at o and whose radius is oc, and let 
the three cones turn in rolling con- 
tact on oc, their axes being fixed ; then 
the point / on the small cone will de- 
scribe a spherical epicycloid fh on the 
spherical base of the cone sct-o pro- 
duced, and a spherical hypocycloid 
fg on the base of the cone o-cu; or 
the element co of the cone o-ce would 
generate the proper surfaces for the 
faces of the teeth on sct-o and for 
flanks on o-uc, these surfaces being 
the same as those formed by allow- 
ing a right line to pass through o and 
move along fh and fg respectively as directrices. As the common normal 
plane to the two tooth surfaces generated by the above method always 
passes through the common element of contact of the two pitch cones, 
viz., oc, therefore they are suitable tooth surfaces, and will maintain by 
their sliding contact the same a.v. ratio as the pitch cones would maintain 
by rolling contact. Since the above method of drawing the shapes of 
bevel gears on a true spherical surface involves much labor, the following 
approximate method, given by Tredgold, is extensively used where 
absolute accuracy is not required. 




Fig. 328. 




Fig. 329. 



Tredgold's Approximation. — In the plane of the axes of the two 
rolling cones (Fig. 328) draw acb perpendicular to oc, intersecting the 



BEVEL GEARING. 



243 



axes in a and b; then cb is an element of the cone b-sct, tangent to the 
sphere at the circle set, and ac is an element of the cone a-cu, tangent 
to the sphere at the circle uc. Since narrow zones of the sphere near 
the circles set and uc will so nearly coincide with cones tangent at these 
circles, the conical surfaces may be substituted for the spherical ones 
without serious error, and, as the tooth outlines are always compara- 
tively short, they may be supposed to lie in the conical surfaces b-sct and 
a-cu, which are called the normal cones, they being normal to the pitch 
cones. These normal cones may now be developed, and the process of 
drawing the tooth outlines will be the same as for a pair of spur gears of 
the required pitch with ac and be as radii. The method of drawing the 
normal cones and obtaining the tooth outlines is shown in Fig. 329, 
which is lettered the same as Fig. 328. 

The demonstrations and methods belonging to the teeth of spur- 




N> 



Fig. 330. 

wheels may be applied to the development of bevel-gear teeth with the 
exception of pin gearing, which requires another system now obsolete. 

Of the various forms of teeth, that is, radial-flank, parallel-flank, 
curved-flank, or involute teeth, the first form — radial-flank — is com- 
monly used. Bevel gears being generally made in pairs, radial flanks 
are preferred as they have the simplest form and give the least obliquity 
of action. The involute tooth is now coming into use for cut bevel 
gears, and very good results have been obtained by its adoption. 



2U CONSTRUCTION OF GEAR-TEETH. 

The method of finding the tooth outline upon the normal cone 
graphically is shown in Fig. 330, where A is an end view, B a side view, 
of the gear-wheel, and C the development of the tooth outline. Since 
the tooth projects beyond the pitch circle to t lt the normal cone will 
extend to t, which fixes the extreme diameter of the " blank" tu; pro- 
jecting r x to at, we find the diameter of the projected root circle rs. The 
points t, c, r, etc., in revolving about the axis ao, describe circles which 
appear as straight lines in B, and as circles in A, projected in their true 
size. It is obvious that the length of the arc which measures the thickness 
of the tooth at the addendum, root, pitch, or any other projected circle 
will be the same in the end view A as in the development, which enables 
us to draw the outlines of the teeth in the end view as shown, all outlines 
being the same. From these outlines those on the side view B may be 
found by the principles of projections. • 

The teeth are limited at their smaller ends by another normal cone 
on which the outline will have the same form as on the large end. Since 
all elements of the teeth run to the vertex o, this second outline may 
be found from the first, the method of obtaining it being sufficiently 
indicated in the figure. W is the development of the tooth outline of 
a wheel to work with the one shown. Both wheels have radial flanks, 
and the development is conveniently located for drawing both wheels at 
the same time; one may be shown in gear with the other. 

Construction of the Correct Tooth Outline. — Let doc (Fig. 331) 
be the pitch cone, fae the normal cone indefinitely extended; and let 
oh be the axis of a describing cone cob, tangent to the pitch cone on 
the line co, and intersecting the normal cone on the curve crstb. The 
right-hand figure is a projection on a plane at right angles to ao, and 
the circle cgd is shown in its true size , c x q x d x ; the curve of intersection 
appears as c 1 r 1 s 1 t 1 b 1 , the method of obtaining two of its points being 
clearly shown. 

Suppose the axes of the pitch and describing cones to be fixed, and 
suppose the cones to turn in rolling contact in the directions shown by 
the arrows; then the element co of the describing cone will sweep up 
the outline for the tooth face, and will always pierce the normal cone 
in some point of the curve ctb, the normal cone being fixed as far as 
its relation to the curve is concerned. Knowing the ratio of the bases 
of the pitch and describing cones, the angular motion of one can be 
found from that of the other, either graphically as shown in the figure, 
or better by calculation. If the pitch cone turns through an angle 
c l a l l = ??ipl, the describing cone will turn through an angle mn\, the 
element will be found at or, giving us the point r t and the small portion 
of the projected tooth outline lr v To find another point, suppose the 
turning to go on to 2, giving the point s u r t having now gone to u, a point 
found on the arc r ± u about a x by making the angle r 1 a 1 u=la l 2. In 



BEVEL GEARING. 



245 



the same way other points may be found, giving the projected outline 
Svivt l which is to be developed, as shown at D, before it can be applied 
to the normal cone to fix the tooth outlines. In order to accurately 
fix the outline, great care must be used and several points should be 
found intermediate to those shown in the figure. 

In laying out the teeth of internal bevel wheels by Tredgold's process, 
it is evident that the size of the describing circles must be fixed with 




Fig. 331. 

due regard to the limits obtained for annular spur-wheels, to prevent 
interference upon the development of the normal cone. From the above 
we deduce, as a safe practical rule for the size of the describing cones 
used in the exact process, that the diameters of their bases should not be 
greater than those of the describing circles used in the approximate 
method. 

Cone and Flat Disc. — When a cone rolls on a flat disc the normal 
cone becomes a cylinder, and if Tredgold's process be applied, a case 
like that of a rack and spur-wheel presents itself. It is to be observed 
that if we start with such a flat bevel-wheel, making its developed rack 
teeth of equal curves in alternate reversion, any two bevel- wheels gearing 
with it will gear with each other (§ 201). 



246 CONSTRUCTION OF GEAR-TEETH. 

Involute Wheels. — Tredgold's process is here applied to the devel- 
oped base circles of the normal cones. The exact outline of the teeth on 
the normal cone would be found by noting on it the intersecting path 
of a line carried by a plane, in rolling contact with two base cones, and 
turning on an axis passing through the apexes of the pitch cones per- 
pendicular to the plane. 

Method of Cutting the Teeth. — The tooth surfaces being conical, 
their outlines are constantly changing; it is then impossible to cut them 
accurately with an ordinary milling- cutter. This method, however, is 
often used for small bevel gears. To distribute the unavoidable errors 
as uniformly as possible, it is the practice to make the cutter conform 
to the middle section of the tooth, and to make it travel on the element 
of the tooth on the pitch cone, where the face and flank join, and at the 
same time along the root cone. The cutter is made narrower than the 
smallest space, and only one side of the tooth is cut at a time. With this 
method the flank of the tooth at the large end is too full and the face 
not full enough; at the small end the errors are reversed; the surfaces 
also are cylindrical and not conical as they should be. Messrs. Brown 
and Sharpe make the clearance the same at both ends of the teeth; the 
cutting angle is thus the complement of the face angle of the gear with 
which the one being cut is to work; they also shape the cutter to be 
correct for a point on the tooth one- third of its breadth from the large end. 

The system of diametral pitch is also applied to small bevel gears, 
the same rules holding, it being always understood that by the pitch 
circle is meant the largest, or that of the base of the rolling pitch cone. 
There are, however, machines which will cut true bevel gears. 

Twisted Bevel-wheels. — It is to be noticed that bevel gears may 
be stepped in the same way as spur gears, and the advantages aris- 
ing would be the same ; but there are practical reasons why this arrange- 
ment is not employed. The wheels may have the process of twisting 
applied to them as in twisted gearing; in such case the only objection is 
the difficulty of forming the teeth: as far as outline goes, any outline that 
is suitable before twisting will also be after twisting. 

206. Screw Gearing. — Worm and Wheel. — The most familiar example 
is that of the Worm and Wheel, where the axes are situated in planes at 
right angles to each other, as shown in Fig. 332. Here let be the centre 
of a pitch circle through c, and tt the pitch line of a rack. In the plane 
of the paper construct teeth on these pitch lines of any proper form for 
spur gearing. If now the rack outline be taken as the meridian section 
of a screw whose pitch is equal to that of the rack, one turn of the screw 
or worm will advance the wheel one tooth, just as though we considered 
the screw to act as a rack, and to be moved along its axis a distance equal 
to the pitch; the wheel being made very thin, the screw action of the 
successive equal meridian sections as they come into the plane of the 



SCREW GEARING. 



247 



paper is the same as that of a uniformly moving rack tooth driving the 
wheel. Hence the screw may be considered as a rack which advances 
by rotation along the axis ss. The line tt, revolving about the axis ss, 




Fig. 332. 
generates the pitch cylinder of the screw ; and this is tangent to the pitch 
cylinder of the wheel at c. 

The action in screw gearing may be distinguished from that of twisted 
gearing by three characteristics. 1° The velocity ratio depends wholly 
upon the screw pitch and not on the relative diameters of the pitch 
cylinders. 2° The directional relation depends upon the twist, one 
motion being given by a right-handed, and the opposite by a left-handed 
screw. 3° The end thrust of the screw causes the motion of the wheel. 

To give the wheel of Fig. 332 sensible thickness, determine the com- 
mon tangent plane of the two pitch cylinders at c, as shown at MN 
(Fig. 333), on which the helices of the screw 
will develop into straight parallel lines, as ab. 
The development of the screw helix passing 
through c will, when the plane MN is wrapped 
upon the pitch cylinder of the wheel, become _r 
another helix lying on that surface; these two 
helices tangent at c will be either both right- 
handed or both left-handed. Now consider 
the helix on the wheel's pitch cylinder as a 
directrix for the wheel section, shown in Fig. 
332, the pitch point c moving along the helix; 

then the section will, in its motion, form a twisted wheel which will 
work with the worm. Here the teeth only touch on points in the cen- 
tral transverse plane and thus the wear is excessive. The thick- 
ness of the wheel depends upon its material and the pressure to be 
transmitted. The teeth of these wheels are sometimes cut straight 
across the wheel with an ordinary milling-cutter, at an angle with the 
elements of the pitch cylinder equal to that between ab and ee. 



m 



Fig. 333. 



248 CONSTRUCTION OF GEAR-TEETH. 

207. Close-fitting Worm and Wheel. — To make such a wheel, an 
exact copy of the screw is made of steel, and then it is fluted and hardened, 
similar to a tap, so as to become a cutting-tool, which may be used to 
finish the teeth, usually roughed out by the method of Fig. 333. Placing 
this cutting- tool in proper position in reference to the axis of the wheel, 
and in the notches previously made, it can be made to cut out the wheel 
by its rotation, the axes being pressed nearer together as the cutting 
goes on. Worm-wheel cutting -machines are now made where the wheel 
can be given the proper rotation in relation to the worm by independent 
mechanism. When the worm is allowed to cut all of the material away, 
no guiding notches being made, the wheel will have more teeth than 
wanted, as the cutting begins on a cylinder larger than the pitch cylinder; 
the tooth form is also unsatisfactory. 

The involute form of tooth is usually applied in worm gearing, as it 
gives a straight-sided screw, and a change in the. distance between the 
axes does not affect the velocity ratio. 

For a close-fitting worm-wheel the blank is usually of the form shown 
at the right in Fig. 332, where the lines oj and o x e through the axis 
of the worm describe cones on the axis oe, which limit the teeth. 

Since all sections of a screw on planes parallel to and equidistant 
from its axis are alike, they will act the same as the meridian section 
of Fig. 332. This enables us to draw the outline for the teeth of a close- 
fitting worm, as shown in Fig. 334, where the view at the left corresponds 
to the section at the right of Fig. 332. The teeth of the wheel follow the 
circle of the worm through an angle 2a, which ought not to exceed 60°. 
The pitch point 0, to secure the strongest tooth on the wheel, should 
be located half-way between / and h, in which case the teeth of the wheel 
will be cut away much less at their points than those shown. Xow 
pass a plane through cd parallel to ab; it will cut from the screw the 
outline of a rack as shown at B, Fig. 334; the conjugate of this rack 
tooth will give the shape of the wheel's tooth on the plane cd. In the 
same way, other planes may be passed parallel to cd. The contour of 
the teeth on the conical sides of the wheel may be found by developing 
the cones and applying a method similar to Tredgold's, used in drawing 
bevel gears. The several sections found must be properly located rel- 
atively to each other, and a sufficient number of outlines will enable the 
wheel pattern to be made. 

It has been found in practice that the worm-wheel, to give good 
results, should not have less than 25 teeth; the obliquity of action for 
an involute wheel tooth may be taken about 15°. 

Hindley Worm. — Fig. 335 shows the close-fitting Hindley worm and 
wheel. Here the contour of the worm corresponds with that of the 
root circle of the wheel at its central plane. The worm is cut with a 
tool shaped to the contour of its thread (in this case straight-sided), but, 



CLOSE-FITTING WORM AND WHEEL. 



249 




250 



CONSTRUCTION OF GEAR-TEETH. 



instead of being advanced uniformly parallel to the axis of the worm, the 

tool is here made to turn uni- 
formly about an axis having 
the same position relative to the 
axis of the worm as the wheel 
to be driven. This angular 
motion for one rotation of the 
worm is the same as that 
desired in the wheel. After 
turning the worm it may be 
made to cut a close-fitting 
wheel in the manner previously 
described for the ordinary close- 
fitting worm and wheel. This 
worm when properly made has 
a greater bearing surface than 
the ordinary form, and hence 
the pressure and wear on the 
teeth of the wheel are both 
distributed and thereby re- 
duced. It is extensively used 
in driving elevator drums. 

Close-fitting worms should 
always be well lubricated, and 
are for that reason usually placed under their wheels, so that they may 
run in a bath of oil, the worm and wheel being enclosed in a suitable 
tight casing. 

Multiple-threaded Screw-wheels. — So far the screw has been sup- 
posed to be single- threaded, its pitch being that of the fundamental 
rack tooth. If now we double the helical pitch, the angular velocity of 
the thin wheel (Fig. 332) will be doubled, and only alternate teeth will 
come into action. To bring the remaining teeth into action, the screw 
can be made double- threaded, and this will at the same time reduce the 
pressure upon each tooth. In the same manner the helical pitch may 
be made any number of times as great as the tooth pitch, the number 
of threads being increased accordingly; the diameter of the screw in 
such case should be made great enough to avoid excessive obliquity of 
action. The screw may then have as many threads as there are teeth 
upon the wheel, or more; the combination will then appear as shown in 
Fig. 273. When the number of teeth on the wheel becomes infinite, the 
wheel becomes a rack, and its teeth will have an outline like a portion 
of an ordinary nut. 

208. Oblique Screw Gearing. — The axis of the screw may cross the 
plane of the wheel obliquely, and give motion to the wheel by its end 




Fig. 335. 



OBLIQUE SCREW GEARING. 



251 




thrust ; the fundamental principle is here the same, the screw being still 
a rack which advances by rotation. 

Oblique Rack and Wheel.— Suppose MN (Fig. 336) to be a broad 
plate with teeth cut across it parallel to ca and 
in gear with the wheel o x . On moving the 
plate in the direction cb, the wheel o x will be 
turned through an angle depending on the com- 
ponent ce of the plane's motion, perpendicular 
to the axis of the wheel, and the sliding of the 
teeth, in a direction parallel to the axis of the 
wheel, will be represented by the component ca. 
The sections of the teeth on a plane perpen- 
dicular to the axis will be the same as in spur 
gearing, and their action will be the same, 
excepting the additional sliding ca. It is easily 
seen that the inclined rack, running between 
suitably shaped guides, will act the same as the 
wide plate. If ce is taken as the pitch arc of Fig. 336. 

the wheel o v then cb will represent the movement of the rack while the 
wheel turns through the pitch angle. 

Oblique Worm and Wheel.— Assume the distance cb (Fig. 336) as 

the helical pitch of an oblique single- 
threaded worm to replace the rack. 
A worm thus constructed is shown in 
Fig. 337, where A is its pitch cylinder, 
m B that of the wheel, and p the point of 
tangency. Let p be the present point 
of contact between a thread and a 
tooth, as shown at p' below, which 
gives an intersection of the wheel and 
worm on the plane Im normal to the 
axis of the wheel. Also let the section 
of the thread have the form of the 
rack tooth of Fig. 336, and make the 
wheel tooth conjugate to it. As the 
screw turns, there will always be a 
section of its thread, like p, similarly 
situated with respect to its axis, travel- 
ling along with uniform speed, as shown by the straight arrow, and 
advancing for one turn of the screw to the position o. To keep the 
velocity ratio constant, this moving section of the thread must always 
act on a tooth outline of the same form. Hence in every normal sec- 
tion of the wheel the teeth will have the same outline, and will be of 
the same length when the outside of the wheel is cylindrical. 




Fig. 337. 



252 



CONSTRUCTION OF GEAR-TEETH. 



The twist of the wheel is here found by the same method as that 
shown in Fig. 333. Developing the helical line through p x (the pitch 
point), as shown at pko (pk being equal to the circumference of A and 
perpendicular to po), we have the angle pok that the common tangent 
tt of the helices in contact at p makes with po. The normal sections 
r' and o' of the wheel, on the planes through r and o, would be the same 
as p' and the wheel would thus be a simple twisted one if made from a 
cylindrical blank. The length of the pitch arc on the wheel is pe, cor- 
responding to ce (Fig. 336), and this must be an aliquot part of the 
pitch circumference. 

Action on Wheel. — It may be seen in Fig. 337 that one turn of the 
worm will drive the wheel through more than the original pitch angle, 

although the pitch of the screw is equal to 
the diagonal pitch of the rack in Fig. 336. 
In that case the rack tooth always acted 
against a surface with rectilinear elements 
perpendicular to the plane of rotation; but 
here the worm acts against helices of the 
same pitch, crossing the plane of rotation 
obliquely. The velocity ratio being con- 
stant, we may confine our attention to the 
helices upon the pitch cylinders, and study 
their action as represented in their devel- 
opments on the common tangent p?ane to 
the pitch surfaces. In Fig. 338 let cc and 
dd be elements of the pitch cylinders of the 
wheel and worm respectively, intersecting at 
p and fixing the tangent plane; also let po 
be the pitch of the worm, pk its pitch circumference, and ok the developed 
helix, as in Fig. 337. tt, parallel to ok, is the common tangent to the two 
helices in contact at p. On one turn of the worm p goes to o, while the 
point p of the wheel must move in the direction pf. If we consider pm 
as the helix of the wheel, and suppose the screw to be pushed along pd, 
acting as a rack, po may be resolved into two components pm and pe. 
The first of these, pm, is simply a sliding component and is ineffective; 
but pe represents the linear motion of the wheel, due to the motion 
po of the screw. Now let pk represent the linear motion of the driving 
point p acting against the helix pm. Its normal and tangential com- 
ponents are pg and pj respectively. The motion of p in the wheel is 
along pf, and its normal component must also be pg, which would again 
give us pe as the linear motion of the wheel. 

Suppose now that a double thread is desired upon the worm, without 
changing the subdivision of the wheel: in such case we must double 
the pitch arc pe, and then pn will be the pitch of the worm, found by 




OBLIQUE SCREW GEARING. 



253 



making pf=2pe, and drawing kfn. t x t x is then the common tangent to the 
two helices, and ph the common normal component. Now in Fig. 338 both 
ho and kn will form right-handed helices upon the pitch 
cylinder of the screw; but on wrapping the tangent 
plane down upon the pitch cylinder of the wheel, kn 
will become a right-handed and ko a left-handed helix. 
Hence there must be an intermediate position in which 
the developed worm helix will be parallel to cc and will 
therefore become a rectilinear element of the wheel's 
pitch cylinder. Such a case is obtained by the pro- 
portions shown in Fig. 339 lettered the same as Fig. 338. 
If pe and the obliquity cpo are given, the pitch and 
circumference of the screw are found by drawing through 
e a perpendicular to pe, cutting po at o, and pk (a per- 
pendicular to po) at k. If pe and pk are given, draw 
an arc about p with a radius pe and also draw po perpendicular to pk; 
then ke tangent to the arc fixes the pitch po and the obliquity cpo. The 
wheel in this case becomes a common spur-wheel, as shown in Fig. 340. 




Fig. 339. 





Fig. 340. 



Fig. 341, 



Oblique Screw and Rack. — If the diameter of the oblique worm- 
wheel be increased indefinitely, it will become a rack whose tooth surfaces 
are made up of rectilinear elements. Such a case is shown in Fig. 341. 

An oblique worm could be made to cut its own wheel just as in the 
common case ; the cylindrical blank may then be made to conform to the 
curvature of the screw, and the teeth be limited by conical frusta instead 
of transverse planes. 



INDEX. 



PAGE 

Accelerated motion , 4 

Addendum 188 

Addendum, limit of, in involute gearing 221 

Addendum line or circle 188 

Aggregate combinations 169-185 

Aggregate motion by linkwork t 169 

Anchor escapement 147 

Angle of action 189 

Angle of obliquity 190, 219 

Angles of approach and recess 189 

Angular velocity 4 

Annular wheels, cycloidal system 209-214 

Annular wheels, involute system 225 

Anti-parallel crank linkage 81-83 

Anti-parallel crank linkage as quick-return motion 82, 83 

Anti-parallel crank linkage, centroids of 82 

Approach, arcs and angles of 189 

Approximate forms of teeth 236-241 

Arc of action 189 

Arcs of approach and recess 189 

Axoid 12 

Back-gears 157 

Backlash 187 

Base circles 219 

Bearings 15 

Bell-crank lever 59, 60 

Belt, effective pull in a 47 

Belt, quarter turn 48-50 

Belts 40 

Belts and pulleys connecting non-parallel axes 47-5^ 

Belts, length of crossed - 42 

Belts, length of open 43 

Bevel gearing 191, 241-246 

255 



256 INDEX. 



PAGE 



Bevel gearing; Tredgold's approximation 242, 243 

Bevel gearing, twisted 246 

Binder pulleys 52 

Boring-bars 181, 182 

Cam 61 

Cam and slotted sliding-bar 144 

Cam, cylindrical 70-72 

Cam, design of, for giving motion on straight line not passing through the 

axis of cam 65 

Cam, design of, for giving motion on straight line passing through the axis 

of cam 62 

Cam diagram for harmonic motion 64 

Cam diagram for uniformly-accelerated and uniformly-retarded motion ... 64 

Cam diagrams 61 

Cam diagrams for giving rapid motions 63 

Cam; general case 66, 67 

Cam, heart 64 

Cam, involute 65 

Cam, positive-motion 68 

Cam, use of roller in a 62 

Centre, instantaneous 9-11 

Centre of motion, periodic 12 

Centroid 12 

Centroids in anti-parallel crank linkage 81, 82 

Chain, Morse rocker-joint 58 

Chain, Reynold silent 57 

Chains 40 

Chains, geared 54-58 

Chains, high-speed 57, 58 

Chronometer escapement 149 

Clearance 188 

Clearing curve 197, 202 

Click or pawl 128 

Clockwork 160 

Closed pair 14 

Closed pair, inversion of 22 

Close-fitting worm and wheel 248 

Cone pulleys 44-47 

Cone pulleys, equal 46 

Cone pulleys solved for crossed belt 44 

Cone pulleys solved for open belt 44, 45 

Cones, Evans friction 29 

Cones, rolling 25-27 

Conic four-bar linkage 109-1 13 

Conjugate curves 194-196 

Cords 40 

Cords and ropes, driving by 52, 53 



INDEX. 257 



PAGE 



Cords, parallel motion by 126 

Cords, small, connecting non-parallel axes 53 

Counter, mechanism of 139, 142 

Crank and rocker 76 

Crossed belt, length of 42 

Crossed belt, solution of cone pulleys using 44 

Crowning of pulleys 41 

Crown-wheel escapement. 146 

Curves, conjugate 194-196 

Curves, rolled 197-202 

Cycloid 197 

Cycloidal gearing 202-219 

Cycloidal gearing; annular wheels 209-214 

Cycloidal gearing; arbitrary proportions 188 

Cycloidal gearing; interchangeable wheels 204 

Cycloidal gearing; low-numbered pinions 214-217 

Cycloidal gearing, path of contact in. . . 204, 205, 207 

Cylinder and sphere, rolling 28 

Cylinder escapement 149 

Cylinders, rolling 23-25 

Cylindrical cam 70-72 

Dead-beat escapement 148 

Dead-points 76 

Describing circles 197 

Describing circles, exterior and interior . 209 

Describing circles, intermediate % 210 

Designing trains, methods of 163-165 

Diagrams for cams 61 

Diagrams for cams giving rapid motions 63 

Differential pulley-block ^ 171 

Differential screws 19 

Disc and roller 28 

Double-acting pawl 132 

Double generation of the epicycloid. 199 

Double generation of the hypocycloid 200 

Double Hooke's joint Ill 

Double oscillation by linkwork , 86 

Double-point gearing 233 

Double rocking-lever 78 

Drag-link 77-79 

Eccentric 98 

Effective pull in a belt 47 

Ellipses, linkage for drawing (elliptic trammel) 105, 106 

Ellipses, linkage for turning (elliptic chuck) 107 

Ellipses, lobed wheels from 37 

Ellipses, rolling 36, 37 



258 INDEX. 



PAGE 



Ellipses, rolling, as quick-return motion 32, 83 

Elliptic chuck 107 

Elliptic trammel 105, 106 

Engine-lathe train 157 

Engine, oscillating 92, 93 

Epicyclic bevel trains. . 178-180 

Epicyclic trains 172-182 

Epicyclic trains used in rope-making 175 

Epicycloid 198 

Epicycloid, double generation of. . . 199 

Epitrochoid, curtate and prolate 201 

Escapement, anchor 147 

Escapement, chronometer .' 149 

Escapement, crown-wheel 146 

Escapement, dead-beat 148 

Escapement, Graham cylinder 149 

Evans friction cones 29 

Expansion of elements in linkwork 97-102 

Exterior describing circle 209 

Feather and groove 16 

Force-closure 22 

Forces, relation between, and linear velocities in mechanism 21 

Forces transmitted by linkwork 84, 85 

Four-bar linkage . 73-86 

Four-bar linkage, conic 109-113 

Four-bar linkage, parallel motion by 125 

Frequency of contact between teeth 161, 162 

Friction-catch 134-137 

Friction-cones, Evans 29 

Friction-gearing 29-31 

Friction-gearing, grooved 31 

Fusee -185 

Gearing 186-253 

Gearing, bevel 191, 241-246 

Gearing-chains 54-58 

Gearing-chains, high-speed 57, 58 

Gearing, friction 29-31 

Gearing, pin. 191, 229-233 

Gearing, screw 192, 246-253 

Gearing, skew 191 

Gearing, spur 190, 202-241 

Gearing, twisted 191 , 235 

Geneva stop 141 

Graham cylinder escapement . 149 

Grant's odontograph 240 

Gravity, motion following law of 64 

Guide-pulleys 48- 



INDEX. 259 

PAGE 

Harmonic motion, diagram for. . 64 

Harmonic motion, linkwork giving 102, 103 

Heart cam 64 

Hindley worm 248 

Hooke's joint 110-113 

Hooke's joint, angular velocity ratio in. . Ill 

Hooke's joint, double Ill 

Hooke's stepped wheels. 235 

Hooke's twisted gearing 235 

Hyperbolas, rolling 38 

Hypocyloid 199, 200 

Hypocycloid, double generation of 200 

Idle wheel 154 

Instantaneous axis 9-11 

Instantaneous axis of rolling bodies 11 

Interchangeable wheels, cycloidal gearing 204 

Interchangeable wheels, involute gearing 228 

Intermediate describing circles 210 

Intermittent linkwork 128-150 

Intermittent motion 140-150 

Interior describing circle. 209 

Involute 200 

Involute cam 65 

Involute gearing 219-229 

Involute gearing ; annular wheels 225 

Involute gearing; arbitrary proportions 229 

Involute gearing; base circles 219 

Involute gearing; interchangeable wheels 228 

Involute gearing; possibility of separating wheels 225-227 

Involute gearing; relation between path of contact and arc of action. . . 221 
Isosceles sliding-block linkage 96, 97 

Klein's odontograph „ 241 

Law governing shapes of tooth curves 192 

Lever 59 

Lever, bell-crank: 59, 60 

Lever, double-rocking 78 

Limacon 233 

Linear velocity 4 

Linear velocity ratio of rigidly-connected points in linkwork 8, 9, 11 

Line of connection in sliding pair 86, 190 

Line of connection in two pulleys connected by a belt 40 

Line of obliquity ' 219 

Linkage 73 

Linkage, angular velocity ratio of cranks in a 74 

Linkage, anti-parallel crank 81-83 

Linkage, conic four-bar 109-113 



260 INDEX. 

PAGE 

Linkage, diagrams to show a. v. or l.v. ratio in a 75 

Linkage, four-bar . 73-86 

Linkage, isosceles sliding-biock 96, 97 

Linkage, sliding-block 87-91 

Linkage, swinging-block 92-94 

Linkage, turning-block 95 

Linkwork, aggregate motion by 169 

Linkwork, double oscillation by 86 

Linkwork, expansion of elements in 97-102 

Linkwork, forces transmitted by. 84, 85 

Linkwork giving harmonic motion : 102, 103 

Linkwork, intermittent 128-150 

Linkwork, slow motion by 84 

Linkwork with one sliding pair 86-102 

Linkwork with two sliding pairs 102-107 

Lobed wheels from ellipses 36, 37 

Lobed wheels from logarithmic spirals 34, 35 

Locking devices 144, 145 

Logarithmic spirals, lobed wheels from 34, 35 

Logarithmic spirals, rolling 32-35 

Machine 1,2 

Mangle-racks . 166 

Mangle-wheels 167 

Masked wheels 137 

Mechanism 2 

Morse rocker-joint chain 58 

Motion, accelerated 4 

Motion, composition and resolution of 7, 8 

Motion, continuous 3 

Motion, intermittent 3 

Motion of translation 12 

Motion, parallelogram of 7, 8 

Motion, parallelopiped of 8 

Motion, periodic centre of 1.2 

Motion, reciprocating 3 

Motion, retarded 4 

Motion, uniformly accelerated and retarded 64 

Mule-pulleys 51 

Non-cylindrical rolling surfaces 30-39 

Non-parallel axes, belts and pulleys connecting 47-52 

Non-parallel axes connected by small cords 53 

Normal pitch 223 

Oblique screw-gearing 250-253 

Oblique worm and wheel 251 

Obliquity, angle of 190, 219 

Obliquity, line of 219 



INDEX. 261 



PAGE 



Obliquity of action. . , 190 

Odontograph, Grant's 240 

Odontography Klein's 241 

Odontograph, Robinson's 240 

Odontograph, Willis' . . . 239 

Odontoids 196 

Oldham's coupling 105 

Open belt, length of 43 

Open belt, solution of cone pulleys using 44, 45 

Oscillating engine 92, 93 

Pairs of elements 14 

Pairs of elements, incomplete. 22 

Pantograph 120-123 

Parabolas, rolling 37 

Parallel cranks 79, 80 

Parallel rod 80 

Parallel motion by cords 126 

Parallel motion by four-bar linkage 125 

Parallel motion; pantograph. 120-123 

Parallel motion, Scott-Russell's 116, 117 

Parallel motion, Watt's 123 

Parallelogram of motion 7, 8 

Parallelopiped of motion 8 

Path of contact 188 

Path of contact and arc of action, relation between, in involute gearing 221-223 

Path of contact in cycloidal gearing 204, 205 

Path of contact in involute gearing 221 

Path of contact, limits of, in cycloidal gearing 207 

Pawl 128 

Pawl, double-acting 132 

Pawl, reversible 130 

Peaucellier's straight-line motion 114 

Periodic centre of motion 12 

Pin gearing 191, 229-233 

Pitch circle or line 151, 186 

Pitch, diametral . . 186 

Pitch-line of a pulley 40 

Pitch, normal 223 

Pitch of a gear-wheel 151, 186 

Pitch of a screw 17 

Pitch point 186 

Pitch surface 186 

Positive-motion cam 68 

Primary and secondary pieces 15 

Pulley-block, differential 171 

Pulley-block, triplex 177 

Pulleys ; angular velocity ratio 41 



262 INDEX. 



PAGE 



Pulleys, belts and, to connect non-parallel axes 47-52 

Pulleys, binder y> 52 

Pulleys, cone 44-47 

Pulleys, crowning of 41 

Pulleys, guide 48 

Pulleys, line of connection of 40 

Pulleys, mule 51 

Pulleys, pitch surface of 40 

Pulleys, stepped 44-47 

Pulleys, tight and loose 42 

Quarter-turn belt 48-50 

Quick-return motion using anti-parallel crank linkage 82, 83 

Quick-return motion using rolling ellipses or elliptic gears 82, 83 

Quick-return motion using swinging-block linkage 93, 94 

Quick-return motion using turning-block linkage 95 

Quick-return motion, Whitworth 95 

Rack and pinion 25 

Rack in involute gearing 224 

Rack, mangle 167 

Ratchet-wheel 128 

Recess, arcs and angles of 189 

Reciprocating motion 3 

Relation between forces and linear velocities in mechanisms 21 

Retarded motion 4 

Reversible pawl 130 

Revolution 3 

Reynold silent chain 57 

Roberts's straight-line motion 124 

Roberts's winding-on motion 182-184 

Robinson's odontograph 240 

Rocker, crank and 76 

Rolled curves 197-202 

Rolling bodies, instantaneous axis of 11 

Rolling cones 25-27 

Rolling contact 23 

Rolling cylinder and sphere 28 

Rolling cylinders 23-25 

Rolling ellipses 36 

Rolling hyperbolas , 38 

Rolling logarithmic spirals 32-35 

Rolling parabolas 37 

Rolling surfaces, non-cylindrical 30-39 

Root line or circle 188 

Ropes, cords and 52, 53 

Ropes, wire 53 

Rotation 3 



INDEX. 263 

PAGE 

Sang's theory 234 

Scott-Russell's parallel motion 116, 117 

Screw and nut 16 

Screw-cutting train 159 

Screw gearing 192, 246-253 

Screw gearing, oblique 250-253 

Screw, pitch of 17 

Screw, power of 21 

Screws, compound 18, 19 

Screws, differential 18, 19 

Screws, methods of cutting 20 

Screws, multiple-threaded 17 

Screws, right-handed and left-handed 17 

Screws, various forms of section of 16 

Sellers feed-discs 30 

Shrouded wheel 233 

Skew gearing 191 

Sliding-block linkage 87-91 

Sliding-block linkage, expansion of elements in 97-101 

Sliding-block linkage, isosceles 96, 97 

Slotted cross-head 102 

Slow motion by linkwork. . 84 

Speed-cones 44-47 

Spiral of Archimedes 65 

Spiral, logarithmic 32-35 

Spiral, logarithmic, lobed wheels from 34, 35 

Spline 16 

Spur gearing 190, 202-241 

Spur gearing; approximate forms of teeth 236-241 

Spur gearing; cycloidal system 202-219 

Spur gearing; involute system 219-229 

Spur gearing; Sang's theory 234 

Spur gearing, twisted 191, 235 

Spur gearing; unsymmetrical teeth ! 234 

Star-wheel 143 

Stepped pulleys 44-47 

Stop-motion, Geneva 141 

Straight-line motion, Peaucellier's 114 

Straight-line motion, Roberts's 124 

Straight-line motion, Scott-Russell's 116, 117 

Straight-line motion, Tchebicheff's 125 

Straight-line motion, Watt's 118, 119, 123 

Sun-and-planet wheels .- 174 

Swash-plate 103 

Swinging-block linkage 92-94 

Swinging-block linkage; quick-return motion 93, 94 

Tchebicheff's straight-line motion 125 

Teeth, frequency of contact between. 161, 162 



264 INDEX. 



PAGET 



Toggle-joint 85 

Tooth-curves, law governing the shapes of 192 

Train, engine-lathe 1 57 

Train of wheels, value of a , ,. 152-154 

Train, screw-cutting 159 

Trains, epicyclic 172-182 

Trains of wheels, directional relation in 153, 154 

Trains of wheels, examples of. . . , 154-161 

Trains of wheels, methods of designing 163-165 

Translation, motion of 12 

Triplex pulley-block 177 

Turning-block linkage 95 

Turning-block linkage ; quick-return motion „ 95 

Twisted bevel gearing 246 

Twisted gearing 191, 235 

Universal joint 112 

Unsymmetrical teeth 234 

Velocity 4 

Velocity, angular 4 

Velocity, linear 4 

Velocity ratio of rigidly-connected points 8, 9, 11 

Watt's parallel motion 123 

Wheels, interchangeable 204, 228 

Wheels in trains 151-168 

Wheels, mangle 166 

Wheels, masked 137 

Whitworth quick return 95 

Whitworth quick return, expansion of 101 

Willis odontograph 239 

Winding-on motion, Roberts's 182-184 

Wiper 1 61 

Wire rope • • • • 53 

Worm and wheel 20, 21, 247 

Worm and wheel, close-fitting 248 

Worm and wheel, Hindley worm 248 

Worm and wheel, oblique 251 

Worm and wheel, velocity ratio in 21 



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AKKANGED UNDER SUBJECTS. 



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1 



5 


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2 


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6 


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Burr's Course on the Stresses in Bridges and Roof Trusses, Arched Ribs, and 

Suspension Bridges 8vo, 3 50 

D« Bois's Mechanics of Engineering. Vol. II Small 4to, 10 00 

Poster's Treatise on Wooden Trestle Bridges 4to, 5 00 

Fowler's Coffer-dam Process for Piers 8vo, 2 50 

Greene's Roof Trusses 8vo, 1 25 

Bridge Trusses 8vo, 2 50 

Arches in Wood, Iron, and Stone 8vo, 2 50 

Howe's Treatise on Arches 8vo, 4 00 

Design of Simple Roof-trusses in Wood and Steel 8vo, 2 00 

Johnson, Bryan, and Turneaure's Theory and Practice in the Designing of 

Modern Framed Structures. Small 4to, 10 00 

Merriman and Tacoby'a Text-book on Roofs and Bridges: 

Part I. — Stresses in Simple Trusses 8vo, 2 50 

Part II.— Graphic Statics 8vo, 2 50 

Part III. — Bridge Design. 4th Edition, Rewritten 8vo, 2 50 

Part IV.— Higher Structures 8vo, 2 50 

Morlson't Memphis Bridge 4*0, 10 00 

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Waddell's De Pontibus, a Pocket-book for Bridge Engineers. . . i6mo, morocco, 3 <* 

Specifications for Steel Bridges nmo, 1 25 

Wood's Treatise on the Theory of the Construction of Bridges and Roofs. 8vo, 2 00 

Wright's Designing of Draw-spans: 

Part L —Plate-girder Draws 8vo, 2 50 

Part II. — Riveted-truss and Pin-connected Long-span Draws 8vo, 2 50 

Two parts in one volume 8vo, 3 50 

HYDRAULICS. 
Basin's Experiments upon the Contraction of the Liquid Vein Issuing from an 

Orifice. (Trautwine.) 8vo, 2 00 

Bovey's Treatise on Hydraulics 8vo, 5 00 

Church's Mechanics of Engineering 8vo, 6 00 

Diagrams of Mean Velocity of Water in Open Channels paper, 1 50 

Coffin's Graphical Solution of Hydraulic Problems i6mo, morocco, 2 50 

Flather's Dynamometers, and the Measurement of Power nmo, 3 00 

Tolwell's Water-supply Engineering 8vo, 4 00 

Prizell's Water-power 8vo, 5 00 

Fuertes's Water and Public Health nmo, 1 50 

Water-filtration Works i2mo, 2 50 

Oanguillet and Kutter's General Formula for the Uniform Flow of Water in 

Rivers and Other Channels. (Her in g and Trautwine.) 8vo, 4 00 

Hazen's Filtration of Public Water-supply 8vo, 3 00 

Hazlehurst's Towers and Tanks for Water- works 8vo, 2 50 

Herschel's 115 Experiments on the Carrying Capacity of Large, Riveted, Metal 

Conduits 8vo, 2 00 

Mason's Water-supply. (Considered Principally from a Sanitary Stand- 
point.) 3d Edition, Rewritten 8vo, 4 00 

Merriman's Treatise on Hydraulics. 9th Edition, Rewritten 8vo, 5 00 

* Michie's Elements of Analytical Mechanics 8vo, 4 00 

Schuyler's Reservoirs for Irrigation, Water-power, and Domestic Water- 
supply Large 8vo, s 00 

•* Thomas and Watt's Improvement of Riyers. (Post., 44 c. additional), 4to, 6 00 

Turneaure and Russell's Public Water-supplies 8vo, 5 00 

Wegmann's Design and Construction of Dams 4to, 5 00 

Water-supply of the City of New York from 1658 to'1895 4to, 10 00 

Weisbach's Hydraulics and Hydraulic Motors. (Du Bois.). 8vo, 5 00 

Wilson's Manual of Irrigation Engineering Small 8vo. 4 00 

Wolff's Windmill as a Prime Mover 8vo, 3 00 

Wood's Turbines. 8vo, 2 50 

Elements of Analytical Mechanics 8vo, 3 00 

MATERIALS OP ENGINEERING. 

Baker's Treatise on Masonry Construction 8vo, 5 00 

Roads and Pavements 8vo, 5 00 

Black's United States Public Works Oblong 4to, 5 00 

Bovey's Strength of Materials and Theory of Structures 8vo, 7 50 

Burr's Elasticity and Resistance of the Materials of Engineering. 6th Edi- 
tion, Rewritten 8vo, 7 50 

Byrne's Highway Construction 8vo, 5 00 

Inspection of the Materials and Workmanship Employed in Construction. 

i6mo, 3 00 

Church's Mechanics of Engineering 8vo, 6 00 

Du Bois's Mechanics of Engineering. Vol. I Small 4to, 7 50 

Johnson's Materials of Construction Large 8vo, 6 00 

Keep's Cast Iron 8vo, 2 50 

Lanza's Applied Mechanics , . . . .8vo, 7 50 

Martens 's Handbook on Testing Materials. (Henning.) 2 vols 8vo, 7 50 

Merrill's Stones for Building and Decoration 8vo, 5 00 

7 



Merriman's Text-book on the Mechanics of Materials 8vo, 

Strength of Materials i2mo, 

Metcalf 's Steel. A Manual for Steel-users iamo, 

Patton's Practical Treatise on Foundations 8vo, 

Richey's Hanbbook for Building Superintendents of Construction. (In press.) 

Rockwell's Roads and Pavements in France i2mo, 

Sabin's Industrial and Artistic Technology of PaintsJand^Varnish 8vo, 

Smith's Materials of Machines i2mo, 

Snow's Principal Species of Wood 8vo, 

Spalding's Hydraulic Cement i2mo, 

Text-book on Roads and Pavements nmo, 

Taylor and Thompson's Treatise on Concrete, PlainfandfReinforced. (In 
press.) 

Thurston's Materials of Engineering. 3 Parts 8vo, 

_ : Part I. — Non-metallic Materials of Engineering and Metallurgy 8vo, 

Part II.— Iron and Steel 8vo, 

Part III. — A Treatise on Brasses, Bronzes, and Other Alloys and their 

Constituents 8vo, 

Thurston's Text-book of the Materials of Construction 8vo, 

Tillson's Street Pavements and Paving Materials 8vo, 

Waddell's De Pontibus. (A Pocket-book for Bridge Engineers.) . . i6mo, mor.. 

Specifications for Steel Bridges i2mo, 

Wood's Treatise on the Resistance of Materials, and an Appendix on the Pres- 
ervation of Timber 8vo, 

Elements of Analytical Mechanics 8vo, 

Wood's Rustless Coatings: Corrosion and Electrolysis of Iron and Steel. . .8vo, 

RAILWAY ENGINEERING. 

Andrews's Handbook for Street Railway Engineers. 3X5 inches, morocco, 

Berg's Buildings and Structures of American Railroads 4to, 

Brooks's Handbook of Street Railroad Location i6mo. morocco, 

Butts's Civil Engineer's Field-book i6mo, morocco, 

Crandall's Transition Curve i6mo, morocco, 

Railway and Other Earthwork Tables 8vo, 

Dawson's "Engineering" and Electric Traction Pocket-book. i6mo, morocco, 
Dredge's History of the Pennsylvania Railroad: (1879) Paper, 

* Drinker's Tunneling, Explosive Compounds, and Rock Drills, 4to, half mor., 

Fisher's Table of Cubic Yards Cardboard, 

Godwin's Railroad Engineers' Field-book and Explorers' Guide i6mo, mor., 

Howard's Transition Curve Field-book i6mo, morocco. 

Hudson's Tables for Calculating the Cubic Contents of Excavations and Em- 
bankments 8vo, 

Molitor and Beard's Manual for Resident Engineers i6mo, 

Nagle's Field Manual for Railroad Engineers i6mo, morocco. 

Philbrick's Field Manual for Engineers i6mo, morocco, 

Searles's Field Engineering i6mo, morocco, 

Railroad Spiral. i6mo, morocco, 

Taylor's Prismoidal Formulae and Earthwork 8vo, 

• Trautwine's Method of Calculating the Cubic Contents of Excavations and 

Embankments by the Aid of Diagrams 8vo, 

The Field Practice of [Laying Out Circular Curves for Railroads. 

i2mo, morocco, 

Cross-section Sheet Paper, 

Webb's Railroad Construction. 2d Edition, Rewritten i6mo. morocco, 

Wellington's Economic Theory of the Location of Railways Small 8vo, 

DRAWING. 
Barr's Kinematics of Machinery 8vo, 2 50 

* Bartlett's Mechanical Drawing 8vo, 3 00 

• " Abridged Ed 8vo, 1 50 

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Coolidge's Manual of Drawing 8vo, paper, i oo 

Coolidge and Freeman's Elements of General Drafting for Mechanical Engi- 
neers. (In preaa.) 

Durley's Kinematics of Machines 8vo, 4 00 

Hill's Text-book on Shades and Shadows, and Perspective 8vo, 2 00 

Jamison's Elements of Mechanical Drawing. (In preaa.) 
Jones's Machine Design: 

Part I. — Kinematics of Machinery 8vo, 1 50 

Part n. — Form, Strength, and Proportions of Parts 8vo, 3 00 

MacCord's Elements of Descriptive Geometr> . 8vo, 300 

Kinematics; or, Practical Mechanism , 8vo, 5 00 

Mechanical Drawing , 4to, 4 00 

Velocity Diagrams 8vo, 1 50 

• Mahan's Descriptive Geometry and Stone-cutting 8vo, 1 so 

Industrial Drawing. (Thompson.) 8vo, 3 50 

Moyer's Descriptive Geometry. (In preaa.) 

Reed's Topographical Drawing and Sketching 4to, 5 00 

Reid's Course in Mechanical Drawing 8vo, 2 00 

Text-book of Mechanical Drawing and Elementary Machine Design. .8vo, 3 00 

Robinson's Principles of Mechanism 8vo, 3 00 

Smith's Manual of Topographical Drawing. (McMillan.) 8vo, 2 50 

Warren's Elements of Plane and Solid Free-hand Geometrical Drawing. . i2mo, 1 00 

Drafting Instruments and Operations nmo, 1 25 

Manual of Elementary Projection Drawing i2mo, 1 50 

Manual of Elementary Problems in the Linear Perspective of Form and : 

Shadow i2mo, x 00 

Plane Problems in Elementary Geometry i2mo, 1 23 

Primary Geometry nmo, 75 

Elements of Descriptive Geometry, Shadows, and Perspective 8vo, 3 50 

General Problems of Shades and Shadows 8vo, 3 00 

Elements of Machine Construction and Drawing 8vo, 7 50 

Problems. Theorems, and Examples in Descriptive Geometry 8vo, 2 50 

Weisbach's Kinematics and the Power of Transmission. (Hermann and 

Klein.) 8vo, 5 00 

Whelpley's Practical Instruction in the Art of Letter Engraving i2mo, 2 00 

Wilson's Topographic Surveying 8vo, 3 50 

Free-hand Perspective 8vo, 2 50 

Free-hand Lettering 8vo, 1 00 

Woolf's Elementary Course in Descriptive Geometry Large 8vo, 3 00 

ELECTRICITY AND PHYSICS. 

Anthony and Brackett's Text-book of Physics. (Magie.) ... .Small 8vo, 3 00 

Anthony's Lecture-notes on the Theory of Electrical Measurements nmo, 1 00 

Benjamin's History of Electricity ; 8vo, 3 00 

Voltaic CelL 8vo, 3 00 

Classen's Quantitative Chemical Analysis by Electrolysis. (Boltwood.). .8vo, 3 00 

Crehore and Squier's Polarizing Photo-chronograph 8vo, 3 00 

Dawson's "Eneineering" and Electric Traction Pocket-book. . i6mo, morocco, 5 00 
Dolezalek's Theory of the Lead Accumulator (Storage Battery). (Von 

Ende.) i2mo," 2 50 

Duhem's Thermodynamics and Chemistry. (Burgess.) 8vo, 4 00 

Flather's Dvnamometers, and the Measurement of Power i2mo, 3 00 

Gilbert's De Magnete. (Mottelay.) 8vo, 2 50 

Hanchett's Alternating Currents Explained i2mo, 1 00 

Hering's Ready Reference Tables (Conversion Factors) i6mo, morocco, 2 50 

Holman's Precision of Measurements 8vo, 2 00 

Telescopic Mirror-scale Method, Adjustments, and Tests Large 8vo, 75 

9 



Landauer's Spectrum Analysis. (Tingle.) 8vo, 3 oa 

Le Chatelier's High-temperature Measurements. (Boudouard — JBurgess.)i2mo, 3 OO 

Lob's Electrolysis and Electrosynthesis of Organic Compounds. (Lorenz.) 1 2mo, z 00 

* Lyons's Treatise on Electromagnetic Phenomena. Vols. I. and II. «vo, each, 6 00 

* Michie. Elements of Wave Motion Relating to Sound and Light 8vo, 4 00 

Niaudet's Elementary Treatise on Electric Batteries. (Fishoack. ) nmo, a 50 

* Rosenberg's Electrical Engineering. (HaldaneGee — Kinzbrunner.) 8vo, 1 50 

Ryan, Norris, and Hozie's Electrical Machinery. VoL L 8vo, a 90 

Thurston's Stationary Steam-engines 8vo, a 50 

* Tillman's Elementary Lessons in Heat 8vo, x 50 

Tory and Pitcher's Manual of Laboratory Physics Small 8vo, a 00 

Ulke's Modern Electrolytic Copper Refining 8vo, 3 00 

LAW. 

* Davis's Elements of Law 8vo, a 50 

* Treatise on the Military Law of United States 8vo, 7 00 

* Sheep, 7 50 

Manual for Courts-martial z6mo, morocco, x 50 

Wait's Engineering and Architectural Jurisprudence 8vo, 6 00 

Sheep, 6 50 
Law of Operations Preliminary to Construction in Engineering and Archi- 
tecture 8vo, 5 oa 

Sheep, 5 50 

Law of Contracts 8vo, 3 oa 

Winthrop's Abridgment of Military Law i2mo, a 5© 

MANUFACTURES. 

Barnadou's Smokeless Powder — Ifitro-cellulose and Theory of the Cellulose 

Molecule xamo, a 5a 

Bolland's Iron Founder i2mo, a s* 

" The Iron Founder," Supplement .- xamo, a 50 

Encyclopedia of Founding and Dictionary of Foundry Terms U6ed in the 

Practice of Moulding xamo, 3 oa 

Slssler's Modern High Explosives 8vo, 4 00 

Bffront's Enzymes and their Applications. (Prescott.) 8vo, 3 00 

Fitzgerald's Boston Machinist i8mo, x 00 

Ford's Boiler Making for Boiler Makers i8mo, x oa 

Hopkins's Oil-chemists' Handbook 8vo, 3 00 

Keep's Cast Iron 8vo, a s« 

Leach's The Inspection and Analysis of Food with Special Reference to State 

Control. (In preparation.) 
Matthews's The Textile Fibres. (In press.) 

Metcalf 's Steel. A Manual for Steel-users nmo, a 00 

Metcalfe's Cost of Manufactures — And the Administration of Workshops, 

Public and Private 8vo, 5 oa 

Meyer's Modern Locomotive Construction 4to, 10 oa 

Morse's Calculations used in Cane-sugar Factories i6mo, morocco, 1 50 

* Reisig's Guide to Piece-dyeing 8vo, as 00 

Sabin's Industrial and Artistic Technology of Paints and Varnish 8vo, 3 00 

Smith's Press-working of Metals 8vo, 3 oa 

Spalding's Hydraulic Cement X2mo, a 00 

Speacar's Handbook for Chemists of Beet-sugar Houses i6mo, morocco, 3 00 

Handbook tor sugar Manufacturers and their Chemists.. . x6mo, morocco, a 00 
Taylor and Thompson's Treatise on Concrete, Plain and Reinforced. (In 

press.) 
Thurston's Manual of Steam-boilers, their Designs, Construction and Opera- 
tion 8vo, 5 00 

10 



* Walke's Lectures on Explosives 8vo, 4 ©• 

West's American Foundry Practice nmo, a 50 

Moulder's Text-book i2mo, a 50 

Wiechmann's Sugar Analysis *. Small 8vo. a 50 

Wolffs Windmill as a Prime Mover 8vo, 3 00 

Woodbury's Fire Protection of Mills 8vo, 2 5« 

Wood's Rustless Coatings: Corrosion and Electrolysis of Iron and Steel. . .8vo, 4 00 

MATHEMATICS. 

Baker's Elliptic Functions 8vo, 1 30 

* Bass's Elements of Differential Calculus nmo, 4 oo 

Briggs's Elements of Plane Analytic Geometry nmo, x 00 

Compton's Manual of Logarithmic Computations nmo, 1 50 

Davis's Introduction to the Logic of Algebra 8vo, 1 50 

* Dickson's College Algebra Large nmo, x 5* 

* Answers to Dickson's College Algebra 8vo, paper, as 

* Introduction to the Theory of Algebraic Equations Large nmo, 1 as 

Halsted's Elements of Geometry 8vo, 1 75 

Elementary Synthetic Geometry 8vo, 1 50 

Rational Geometry nmo, 

* Johnson's Three-place Logarithmic Tables: Vest-pocket size paper, 15 

100 copies for 5 00 

* Mounted on heavy cardboard, 8 X 10 inches, as 

xo copies for a 00 

Elementary Treatise on the Integral Calculus Small 8vo, 1 50 

Curve Tracing in Cartesian Co-ordinates nmo, 1 00 

Treatise on Ordinary and Partial Differential Equations Small 8vo, 3 50 

Theory of Errors and the Method of Least Squares nmo, 1 so 

* Theoretical Mechanics nmo, 3 00 

Laplace's Philosophical Essay on Probabilities. (Truscott and Emory.) nmo, a 00 

* Ludlow and Bass. Elements of Trigonometry and Logarithmic and Other 

Tables. 8vo, 3 00 

Trigonometry and Tables published separately ,. Each, a 00 

* Ludlow's Logarithmic and Trigonometric Tables 8vo, x 00 

Maurer's Technical Mechanics. 8vo, 4 00 

Merriman and Woodward's Higher Mathematics 8vo, 5 00 

Merriman's Method of Least Squares 8vo, a 00 

Rice and Johnson's Elementary Treatise on the Differential Calculus. Sm., 8vo, 3 00 

Differential and Integral Calculus, a vols, in one Small 8vo, 2 50 

Sabin's Industrial and Artistic Technology of Paints and Varnish 8vo, 3 00 

Wood's Elements of Co-ordinate Geometry 8vo, a 00 

Trigonometry: Analytical, Plane, and Spherical 12 mo, 1 00 

MECHANICAL ENGINEERING. 

MATERIALS OF ENGINEERING, STEAM-ENGINES AND BOILERS. 

Bacon's Forge Practice nmo, 1 50 

Baldwin's Steam Heating for Buildings nmo, 2 50 

Barr's Kinematics of Machinery 8vo, 2 50 

* Bartlett's Mechanical Drawing 8vo, 3 00 

* " " " Abridged Ed 8vo, 1 s« 

Benjamin's Wrinkles and Recipes nmo, 3 00 

Carpenter's Experimental Engineering 8vo, 6 00 

Heating and Ventilating Buildings 8vo, 4 oo 

Cary's Smoke Suppression in Plants using Bituminous Coal. {In prej>- 
aration.) 

Clerk's Gas and Oil Engine Small 8vo, 4 00 

Coolidge's Manual of Drawing 8vo, paper, 1 00 

11 



Coolidge and Freeman's Elements of General Drafting for Mechanical En- 
gineers. (In vress.) 
Cromwell's Treatise on Toothed Gearing i2mo, 

Treatise on Belts and Pulleys i2mo» 

Durley's Kinematics of Machines * 8vo, 

Flather's Dynamometers and the Measurement of Power 12 mo. 

Rope Driving 12020, 

Gill's Gas and Fuel Analysis for Engineers i2mo, 

Hall's Car Lubrication nmo, 

Hering's Ready Reference Tables (Conversion Factors) i6mo, morocco, 

Hutton's The Gas Engine 8vo, 

Jones's Machine Design: 

Part I. — Kinematics of Machinery 8vo, 

Part II. — Form, Strength, and Proportions of Parts 8vo, 

Kent's Mechanical Engineer's Pocket-book i6mo, morocco, 

Kerr's Power and Power Transmission 8vo, 

Leonard's Machine Shops, Tools, and Methods. (In press.) 

MacCord's Kinematics ; or, Practical Mechanism 8vo, 

Mechanical Drawing 4to, 

Velocity Diagrams 8vo, 

Mahan's Industrial Drawing. (Thompson.) 8vo, 

Poole's Calorific Power of Fuels 8vo, 

Reid's Course in Mechanical Drawing 8vo. 

Text-book of Mechanical Drawing and Elementary Machine Design. .8vo, 

Richards's Compressed Air i2mo, 

Robinson's Principles of Mechanism 8vo, 

Schwamb and Merrill's Elements of Mechanism. (In press.) 

Smith's Press-working of Metals 8vo, 

Thurston's Treatise on Friction and Lost Work in Machinery and Mill 
Work 8vo, 

Animal as a Machine and Prime Motor, and the Laws of Energetics . umo, 

Warren's Elements of Machine Construction and Drawing 870, 

Weisbach's Kinematics and the Power of Transmission. Herrmann — 

Klein.) 8vo, 

Machinery of Transmission and Governors. (Herrmann- — Klein.). .8vo, 

HydrauLcs and Hydraulic Motors. (Du Bois. ) 8vo, 

Wolff's Windmill as a Prime Mover 8vo, 

Wood's Turbines 8vo, 

MATERIALS OF ENGINEERING. 

Bovey's Strength of Materials and Theory of Structures 8vo, 7 5<» 

Burr's Elasticity and Resistance of the Materials of Engineering. 6th Edition, 

Reset 8vo. 7 50 

Church's Mechanics of Engineering 8vo, 6 00 

Johnson'* Materials of Construction Large 8vo, 6 00 

Keep's Cast Iron 8vo, 2 50 

Lanza's Applied Mechanics 8vo, 7 50 

Martens's Handbook on Testing Materials. (Henning.) 8vo, 7 SO 

Merriman's Text-book on the Mechanics of Materials 8vo, 4 00 

Strength of Mater»als i2mo, 1 00 

Metcalf's Steel A Manual for Steel-users i2mo. 2 do 

Sabin's Industrial and Artistic Technology of Paints and Varnish 8vo, 3 00 

Smith's Materials of Machines i2mo, 1 00 

Thurston's Materials of Engineering 3 vols , Svo, 8 00 

Part H.— Iron and Steel 8vo, 3 50 

Part HI. — A Treatise on Brasses, Bronzes, and Other Alloys and their 

Constituents 8vo 2 50 

Text-book of the Materials of Construction 870 , 5 00 

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Wood's Treatise on the Resistance of Materials and an Appendix on the 

Preservation of Timber 8vo, 2 00 

Elements of Analytical Mechanics 8vo, 3 



00 



Wood's Rustless Coatings: Corrosion and Electrolysis of Iron and Steel. . .8vo, 4 00 



STEAM-ENGINES AND BOILERS. 

Carnot's Reflections on the Motive Power of Heat. (Thurston.) i2mo, 1 50 

Dawson's "Engineering" and Electric Traction Pocket-book. .T6mo, rnor., 5 00 

Ford's Boiler Making for Boiler Makers i8mo, 1 00 

Goss's Locomotive Sparks . 8vo, 2 00 

Hemenway's Indicator Practice and Steam-engine Economy , i2mo, 2 00 

Hutton's Mechanical Engineering of Power Plants 8vo, 5 00 

Heat and Heat-engines 8vo, 5 00 

Kent's Steam-boiler Economy 8vo, 4 00 

Kneass's Practice and Theory of the Injector 8vo 1 50 

MacCord's Slide-valves 8vo, 2 00 

Meyer's Modern Locomotive Construction 4to t 10 00 

Peabody's Manual of the Steam-engine Indicator 12 mo, z 50 

Tables of the Properties of Saturated Steam and Other Vapors 8vo, 1 00 

Thermodynamics of the Steam-engine and Other Heat-engines 8vo, 5 00 

Valve-gears for Steam-engines 8vo, 2 50 

Peabody and Miller's Steam-boilers 8vo, 4 00 

Pray's Twenty Years with the Indicator Large 8vo, 2 50 

Pupln's Thermodynamics of Reversible Cycles in Gases and Saturated Vapors. 

(Osterberg.) nmo, 1 25 

Reagan's Locomotives : Simple, Compound, and Electric nmo, 2 50 

Rontgen's Principles of Thermodynamics. (Du Bois.) 8vo, 5 00 

Sinclair's Locomotive Engine Running and Management nmo, 2 00 

Smart's Handbook of Engineering Laboratory Practice nmo, 2 50 

Snow's Steam-boiler Practice 8vo, 3 00 

Spangler's Valve-gears 8vo, 2 50 

Notes on Thermodynamics i2mo, 1 00 

Spangler, Greene, and Marshall's Elements of Steam-engineering 8vo, 3 00 

Thurston's Handy Tables 8vo, 1 50 

Manual of the Steam-engine 2 vols. 8vo, 10 00 

Part I. — History, Structuce, and Theory 8vo, 6 00 

Part H. — Design, Construction, and Operation 8vo, 6 00 

Handbook of Engine and Boiler Trials, and the Use of the Indicator and 

the Prony Brake 8vo 5 00 

Stationary Steam-engines 8vo, 2 50 

Steam-boiler Explosions in Theory and in Practice 1 2 mo 1 50 

Manual of Steam-boilers , Their Designs, Construction , and Operation . 8 vo , 5 00 

Weisbach's Heat, Steam, and Steam-engines. (Du Bois.) 8vo, 5 00 

Whitham's Steam-engine Design 8vo, 5 00 

Wilson's Treatise on Steam-boilers. (Flather.) i6mo, 2 50 

Wood's Thermodynamics Heat Motors, and Refrigerating Machines 8vo, 4 00 



MECHANICS AND MACHINERY. 

Barr's Kinematics of Machinery 8vo, 2 50 

Bovey's Strength of Materials and Theory of Structures 8vo, 7 50 

Chase's The Art of Pattern-making i2mo, 2 50 

Chordal. — Extracts from Letters i2mo, 2 00 

Church's Mechanics of Engineering 8vo, 6 00 

Notes and Examples in Mechanics 8vo, 2 00 

13 . 



Compton's First Lessons in Metal-working zamo, 

Compton and De Groodt's The Speed Lathe iamo, 

Cromwell's Treatise on Toothed Gearing i2mo, 

Treatise on Belts and Pulleys umo, 

Dana's Text-book of Elementary Mechanics for the Use of Colleges and 

Schools zamo, 

Dingey's Machinery Pattern Making zamo, 

Dredge's Record of the Transportation Exhibits Building of the World's 

Columbian Exposition of 1893 4to, half morocco, 

Du Bois'i Elementary Principles of Mechanics: 

VoL I. — Kinematics 8vo, 

Vol. II. — Statics 8vo, 

Vol. LTI.— Kinetics 8vo, 

Mechanics of Engineering. VoL I Small 4to, 

VoLIL. Small 4to, 

Duxley's Kinematics of Machines 8vo, 

Fitzgerald's Boston Machinist i6mo, 

Flather's Dynamometers, and the Measurement of Power zamo, 

Rope Driving iamo, 

Gom's Locomotive Sparks 8vo 

Hail's Car Lubrication iamo, 

Holly's Art of Saw Filing i8mo. 

* Johnson's Theoretical Mechanics iamo, 

Statics by Graphic and Algebraic Methods 8vo, 

Jones's Machine Design: 

Part I. — Kinematics of Machinery 8vo, 

Part II. — Form, Strength, and Proportions of Parts 8vo, 

Kerr's Power and Power Transmission 8vo, 

Lanza's Applied Mechanics 8vo, 

Leonard s Machine Shops, Tools, and Methods. (In preaa.) 

MacCord's Kinematics; or, Practical Mechanism 8vo, 

Velocity Diagrams 8vo, 

Maurer's Technical Mechanics 8vo, 

Merriman's Text-book on the Mechanics of Material* 8to, 

• Michie's Elements of Analytical Mechanics 8vo, 

Reagan's Locomotives: Simple. Compound, and Electric tamo, 

Reid's Course in Mechanical Drawing 8vo, 

Text-book of Mechanical Drawing and Elementary Machine Design. .8vo, 

Richard&'s Compressed Air zamo, 

Robinson's Principles of Mechanism 8vo, 

Ryan, Norris, and Hoxie'a Electrical Machinery. Vol. I 8vo, 

Schwamb and Merrill's Elements of Mechanism. (In press.) 

Sinclair's Locomotive-engine Running and Management zamo, 

Smith's Press-working of Metals 8vo, 

Materials of Machines zamo, 

Spangler, Greene, and Marshall's Elements of Steam-engineering 8vo, 

Thurston's Treatise on Friction and Lost Work in Machinery and Mill 
Work 8vo, 

Animal as a Machine and Prime Motor, and the Laws of Energetics, zamo, 

Warren's Elements of Machine Construction and Drawing 8vo, 

Weisbach's Kinematics and the Power of Transmission. (Herrmann — 
Klein.) 8vo, 

Machinery of Transmission and Governors. (Herrmann — Klein.). 8 vo, 
Wood's Elements of Analytical Mechanics 8vo, 

Principles of Elementary Mechanics Z2mo, 

Turbines 8vo, 

The World's Columbian Exposition of 1893 4to, 

14 



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METALLURGY. 

Egleston's Metallurgy of Silver, Gold, and Mercury: 

VoL I.— Silver 8vo, 7 So 

Vol. n. — Gold and Mercury 8vo, 7 5© 

** Iles's Lead-smelting. (Postage o cents additional) iamo, a 50 

Keep's Cast Iron 8vo, a 50 

Kunhardt's Practice of Ore Dressing in Europe 8vo, i 50 

Le Chatelier's High-temperature Measurements. (Boudouard — Burgess.) . iamo, 3 00 

Metcalf' s SteeL A Manual for Steel-users iamo, a 00 

Smith's Materials of Machines iamo, 1 00 

Thurston's Materials of Engineering. In Three Parts 8vo, 8 00 

Part n. — Iron and Steel 8vo, 3 50 

Part III. — A Treatise on Brasses, Bronzes, and Other Alloys and their 

Constituents 8vo, a 50 

Ulke's Modern Electrolytic Copper Refining 8vo, 3 00 

MINERALOGY. 

Barringer's Description of Minerals of Commercial Value. Oblong, morocco, a 50 

Boyd's Resources of Southwest Virginia 8vo, 3 00 

Map of Southwest Virginia , Pocket-book form, a 00 

Brush's Manual of Determinative Mineralogy. (Penfield.) 8vo, 4 00 

Chester's Catalogue of Minerals 8vo, paper, z 00 

Cloth, z as 

Dictionary of the Names of Minerals 8vo, 3 50 

Dana's System of Mineralogy Large 8vo, half leather, xa 50 

First Appendix to Dana's New "System of Mineralogy.". ...Large 8vo, 1 00 

Text-book of Mineralogy 8vo, 4 00 

Minerals and How to Study Them. iamo, 1 90 

Catalogue of American Localities of Minerals Large 8ro, 1 00 

Manual of Mineralogy and Petrography xamo, a 00 

Eakle's Mineral Tables. 8vo, 1 as 

Egleston's Catalogue of Minerals and Synonyms 8vo, a so 

Hussak's The Determination of Rock-forming Minerals. (Smith.) Small 8ro, a 00 

Merrill's Non-metallic Minerals: Their Occurrence and Uses. 8ro, 4 00 

* Penfield's Notes on Determinative Mineralogy and Record of Mineral Tests. 

8vo, paper, o 30 
Rosenbusch's Microscopical Physiography of the Rock-making Minerals. 

(ladings.) 8vo, 5 00 

* Tillman's Text-book of Important Minerals and Docks 8vo, a 00 

Williams's Manual of Lithology 8vo, 3 00 

MINING. 

Beard's Ventilation of Mines xamo, a 50 

Boyd's Resources of Southwest Virginia 8vo, 3 00 

Map of Southwest Virginia Pocket-book form, a 00 

* Drinker's Tunneling, Explosive Compounds, and Rock Drills. 

4to, half morocco, as 00 

Bissler's Modern High Explosives * 8vo, 4 00 

Fowler's Sewage Works Analyses iamo, a 00 

Goodyear's Coal-mines of the Western Coast of the United States xamo, a 50 

Dilseng's Manual of Mining 8vo, 4 00 

•• Iles's Lead-smelting. (Postage 9c. additional.) xamo, a 50 

Kunhardt's Practice of Ore Dressing in Europe 8vo, x 50 

O'Driscoll's Notes on the Treatment of Gold Ores 8vo, a 00 

* Walke's Lectures on Explosives 8vo, 4 00 

Wilson's Cyanide Processes xamo, x 50 

Chlorination Process xamo, 1 50 

Hydraulic and Placer Mining iamo, a 00 

Treatise on Practical and Theoretical Mine Ventilation iamo 1 as 

15 



SANITARY SCIENCE. 

Copeland's Manual of Bacteriology. (In preparation.) 

Folwell's Sewerage. (Designing, Construction and Maintenance.; 8vo, 3 00 

Water-supply Engineering 8vo, 4 00 

Fuertes's Water and Public Health umo , x 50 

Water-filtration Works nmo, 2 50 

Gerhard's Guide to Sanitary House-inspection x6mo, x 00 

Goodrich's Economical Disposal of Town's Refuse Demy 8vo, 3 50 

Hazen's Filtration of Public Water-supplies 8vo, 3 00 

Kiersted's Sewage Disposal . nmo, x 25 

Leach's The Inspection and Analysis of Food with Special Reference to State 

Control. (In preparation.) 
Mason's Water-supply. (Considered Principally from a Sanitary Stand- 
point.) 3d Edition, Rewritten 8vo, 4 o« 

Examination of Water. (Chemical and Bacteriological) nmo, 1 25 

Merriman's Elements of Sanitary Engineering 8vo, 2 00 

Nichols's Water-supply. (Considered Mainly from a Chemical and Sanitary 

Standpoint.) (1883.) * 8vo, 2 so 

Ogden's Sewer Design nmo, 2 00 

Prescott and Winslow's Elements of Water Bacteriology , with Special Reference 

to Sanitary Water Analysis nmo, 1 25 

* Price's Handbook on Sanitation nmo, 1 50 

Richards's Cost of Food. A Study in Dietaries nmo, 1 00 

Cost of Living as Modified by Sanitary Science nmo, 1 00 

Richards and Woodman's Air, Water, and Food from a Sanitary Stand- 
point 8vo, 2 00 

* Richards and Williams's The Dietary Computer 8vo, x 50 

Rideal's Sewage and Bacterial Purification of Sewage 8vo, 3 50 

Turneaure and Russell's Public Water-supplies 8vo, 5 00 

Whipple's Microscopy of Drinking-water 8vo, 3 50 

Woodhull's Notes and Military Hygiene x6mo, 1 50 

MISCELLANEOUS. 

Barker's Deep-sea Soundings 8vo, 2 00 

Emmons's Geological Guide-book of the Rocky Mountain Excursion of the 

International Congress of Geologists Large 8vo x 50 

Ferrel's Popular Treatise on the Winds , 8vo 4 00 

Haines's American Railway Management nmo, 2 50 

Mott's Composition, Digestibility, and Nutritive Value of Food. Mounted chart. 1 25 

Fallacy of the Present Theory of Sound i6mo 1 00 

Ricketts's History of Rensselaer Polytechnic Institute, 1824-1894. Small 8vo, 3 00 

Rotherham's Empnasized New Testament Large 8vo, 2 00 

Steel's Treatise on the Diseases of the Dog 8vo, 3 50 

Totten's Important Question in Metrology 8vo 2 50 

The World's Columbian Exposition ot 1893 4to, x 00 

Von Behring's Suppression of Tuberculosis. (Bolduan.) (In press.) 
Worcester and Atkinson. Small Hospitals, Establishment and Maintenance, 
and Suggestions for Hospital Architecture, with Plans for a Small 

Hospital nmo, t 25 

HEBREW AND CHALDEE TEXT-BOOKS. 

Green's Grammar of the Hebrew Language 8vo, 3 00 

Elementary Hebrew Grammar nmo, x 25 

Hebrew Chrestomathy 8vo, 2 00 

Gesenius's Hebrew and Chaldee Lexicon to the Old Testament Scriptures. 

(Tregelles.) Small 4to, half morocco, 5 00 

Letteris's Hebrew Bible 8vo, 2 2 

16 



NOV 4 1904 i 



